Chapter 11 Exponents & Powers

Given:

The target value is $\frac{-8}{27}$.

The given options are:

(a) $-\left( \frac{2}{3} \right)^3$

(b) $\left( \frac{-2}{3} \right)^3$

(c) $-\left( \frac{-2}{3} \right)^3$

(d) $\left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right)$

To Find:

The number which is not equal to $\frac{-8}{27}$ among the given options.

Solution:

We evaluate each option to see if it is equal to $\frac{-8}{27}$.

Recall that $8 = 2^3$ and $27 = 3^3$. Thus, $\frac{8}{27} = \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^3$. Also, $\frac{-8}{27} = -\frac{8}{27} = -\left(\frac{2}{3}\right)^3$, and $\frac{-8}{27} = \frac{(-2)^3}{3^3} = \left(\frac{-2}{3}\right)^3$.

Let's check each option:

(a) $-\left( \frac{2}{3} \right)^3 = - \frac{2^3}{3^3} = - \frac{8}{27}$.

This is equal to the target value $\frac{-8}{27}$.

(b) $\left( \frac{-2}{3} \right)^3 = \frac{(-2)^3}{3^3} = \frac{-8}{27}$.

This is equal to the target value $\frac{-8}{27}$.

(c) $-\left( \frac{-2}{3} \right)^3 = - \left( \frac{(-2)^3}{3^3} \right) = - \left( \frac{-8}{27} \right)$.

When we multiply a negative number by $-1$, the result is positive.

$- \left( \frac{-8}{27} \right) = \frac{8}{27}$.

This is not equal to the target value $\frac{-8}{27}$.

(d) $\left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right)$ is the definition of $\left( \frac{-2}{3} \right)^3$.

So, $\left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right) \times \left( \frac{-2}{3} \right) = \left( \frac{-2}{3} \right)^3 = \frac{(-2)^3}{3^3} = \frac{-8}{27}$.

This is equal to the target value $\frac{-8}{27}$.

Comparing the results, we see that options (a), (b), and (d) are all equal to $\frac{-8}{27}$, while option (c) is equal to $\frac{8}{27}$.

Conclusion:

The number which is not equal to $\frac{-8}{27}$ is given by option (c).

Given:

The expression $(-7)^5 \times (-7)^3$.

The options are:

(a) $(-7)^8$

(b) $-(7)^8$

(c) $(-7)^{15}$

(d) $(-7)^2$

To Find:

The option that is equal to the given expression.

Solution:

We need to simplify the expression $(-7)^5 \times (-7)^3$.

We can use the rule of exponents for multiplication with the same base, which states that for any non-zero base $a$ and integers $m$ and $n$:

$a^m \times a^n = a^{m+n}$

In this expression, the base is $a = -7$, and the exponents are $m = 5$ and $n = 3$.

Applying the rule:

Now, we add the exponents:

$5 + 3 = 8$

So, the expression simplifies to:

$(-7)^{5+3} = (-7)^8$

Now let's compare this result with the given options:

(a) $(-7)^8$

(b) $-(7)^8$

(c) $(-7)^{15}$

(d) $(-7)^2$

Our simplified expression $(-7)^8$ matches option (a).

Note that option (b) $-(7)^8$ is equivalent to $-1 \times 7^8 = -(7^8)$, which is different from $(-7)^8$. When the base is negative and the exponent is even, the result is positive, i.e., $(-7)^8 = (7^8)$. So, $(-7)^8 = 7^8$, while $-(7)^8 = -7^8$.

Conclusion:

The expression $(-7)^5 \times (-7)^3$ is equal to $(-7)^8$. Therefore, the correct option is (a).

Given:

The expression $x^3 \div y^3$, where $x$ and $y$ are non-zero integers.

The options are:

(a) $\left( \frac{x}{y} \right)^0$

(b) $\left( \frac{x}{y} \right)^3$

(c) $\left( \frac{x}{y} \right)^6$

(d) $\left( \frac{x}{y} \right)^9$

To Find:

The expression that is equal to $x^3 \div y^3$ among the given options.

Solution:

We are asked to simplify the expression $x^3 \div y^3$.

Division can be written as a fraction:

$x^3 \div y^3 = \frac{x^3}{y^3}$

We can use the property of exponents that states for any non-zero numbers $a$ and $b$, and any integer $m$:

$\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$

In our expression, $a = x$, $b = y$, and $m = 3$. Since $x$ and $y$ are non-zero integers, this property is applicable.

Applying the rule to $\frac{x^3}{y^3}$:

So, $x^3 \div y^3$ is equal to $\left(\frac{x}{y}\right)^3$.

Now, let's compare this result with the given options:

(a) $\left( \frac{x}{y} \right)^0$: Any non-zero number raised to the power of 0 is 1. Since $x$ and $y$ are non-zero, $\frac{x}{y}$ is non-zero. Thus, $\left( \frac{x}{y} \right)^0 = 1$. This is generally not equal to $\left(\frac{x}{y}\right)^3$ unless $\left(\frac{x}{y}\right)^3 = 1$, which means $\frac{x}{y} = 1$ (i.e., $x=y$).

(b) $\left( \frac{x}{y} \right)^3$: This matches our simplified expression.

(c) $\left( \frac{x}{y} \right)^6$: This is not equal to $\left(\frac{x}{y}\right)^3$ unless $\left(\frac{x}{y}\right)^3 = \left(\frac{x}{y}\right)^6$. If $\frac{x}{y} \neq 0, 1, -1$, this is only possible if $3=6$, which is false. If $\frac{x}{y}=0$, not applicable as $x, y$ are non-zero. If $\frac{x}{y}=1$, $1^3=1^6=1$. If $\frac{x}{y}=-1$, $(-1)^3=-1$ and $(-1)^6=1$, so $-1 \neq 1$. Thus, it's generally not equal.

(d) $\left( \frac{x}{y} \right)^9$: Similar to option (c), this is generally not equal to $\left(\frac{x}{y}\right)^3$.

The expression $x^3 \div y^3$ is always equal to $\left(\frac{x}{y}\right)^3$ for non-zero integers $x$ and $y$, based on the properties of exponents.

Conclusion:

The expression $x^3 \div y^3$ is equal to $\left(\frac{x}{y}\right)^3$. Therefore, the correct option is (b).

Given:

The expression $(5^7 \div 5^6)^2$.

To Find:

The value of the given expression.

Solution:

We need to evaluate the expression $(5^7 \div 5^6)^2$.

First, we simplify the expression inside the parenthesis, which is $5^7 \div 5^6$.

We use the rule for dividing powers with the same base: $a^m \div a^n = a^{m-n}$, where $a$ is a non-zero base and $m, n$ are integers.

Here, the base is $a=5$, the exponent in the numerator is $m=7$, and the exponent in the denominator is $n=6$.

Calculate the difference in exponents:

$7 - 6 = 1$

So, the expression inside the parenthesis simplifies to:

$5^{7-6} = 5^1 = 5$

Now, we raise this result to the power of 2, as indicated by the outer exponent in the original expression $(5^7 \div 5^6)^2$.

So, $(5^7 \div 5^6)^2 = (5)^2$.

Now, we calculate $5^2$:

$5^2 = 5 \times 5 = 25$

Alternatively, we can use the rule $(a \div b)^m = a^m \div b^m$ and $(a^m)^n = a^{mn}$.

$(5^7 \div 5^6)^2 = (5^7)^2 \div (5^6)^2$

$(5^7)^2 = 5^{7 \times 2} = 5^{14}$

$(5^6)^2 = 5^{6 \times 2} = 5^{12}$

So, the expression becomes $5^{14} \div 5^{12}$.

Using the division rule again:

$5^{14} \div 5^{12} = 5^{14-12} = 5^2 = 25$

Both methods yield the same result.

Answer:

$(5^7 \div 5^6)^2 = \underline{25}$.

Given:

The expression $\frac{a^7b^3}{a^5b}$.

We assume $a \neq 0$ and $b \neq 0$ for the expression to be defined.

To Find:

The simplified form of the expression $\frac{a^7b^3}{a^5b}$.

Solution:

We need to simplify the expression $\frac{a^7b^3}{a^5b}$.

We can rewrite the expression by separating the terms with the same base:

$\frac{a^7b^3}{a^5b} = \frac{a^7}{a^5} \times \frac{b^3}{b}$

We use the rule of exponents for division with the same base, which states that for any non-zero base $x$ and integers $m$ and $n$, $\frac{x^m}{x^n} = x^{m-n}$.

Applying this rule to the terms with base $a$ ($x=a$, $m=7$, $n=5$):

Applying this rule to the terms with base $b$ ($x=b$, $m=3$, $n=1$, since $b = b^1$):

Now, we perform the subtraction in the exponents:

$7 - 5 = 2$

$3 - 1 = 2$

So, the expression simplifies to:

$a^2 \times b^2$

This can also be written as $(ab)^2$ using the rule $(xy)^m = x^m y^m$.

Answer:

$\frac{a^7b^3}{a^5b} = \underline{a^2b^2}$.

Given:

The statement: In the number $7^5$, 5 is the base and 7 is the exponent.

To State:

Whether the given statement is True or False.

Solution:

An exponential expression is written in the form $a^b$, where $a$ is called the base and $b$ is called the exponent (or power).

The base is the number being multiplied by itself, and the exponent indicates how many times the base is multiplied by itself.

In the given number $7^5$:

• The number being multiplied by itself is $7$. This is the base.
• The number of times the base is multiplied by itself is $5$. This is the exponent.

So, in the expression $7^5$, the base is $7$ and the exponent is $5$.

The given statement says that 5 is the base and 7 is the exponent. This contradicts our finding.

Conclusion:

The statement "In the number $7^5$, 5 is the base and 7 is the exponent" is False.

Given:

The statement: $\frac{a^4}{b^3} = \frac{a + a + a + a}{b + b + b}$.

We assume $a$ and $b$ are such that the denominators are non-zero (i.e., $b \neq 0$).

To State:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the given equation and see if they are equal.

Let's look at the left side of the equation: $\frac{a^4}{b^3}$.

By definition of exponents, $a^4$ means $a$ multiplied by itself 4 times:

$a^4 = a \times a \times a \times a$

And $b^3$ means $b$ multiplied by itself 3 times:

$b^3 = b \times b \times b$

So, the left side is:

$\frac{a^4}{b^3} = \frac{a \times a \times a \times a}{b \times b \times b}$

Now, let's look at the right side of the equation: $\frac{a + a + a + a}{b + b + b}$.

The numerator $a + a + a + a$ represents the sum of $a$ added to itself 4 times, which is equivalent to multiplication:

$a + a + a + a = 4 \times a = 4a$

The denominator $b + b + b$ represents the sum of $b$ added to itself 3 times, which is equivalent to multiplication:

$b + b + b = 3 \times b = 3b$

So, the right side is:

$\frac{a + a + a + a}{b + b + b} = \frac{4a}{3b}$

Now, we compare the simplified forms of both sides:

Left side: $\frac{a \times a \times a \times a}{b \times b \times b}$

Right side: $\frac{4a}{3b}$

These two expressions are generally not equal. For example, if $a=2$ and $b=2$:

Left side: $\frac{2^4}{2^3} = \frac{16}{8} = 2$.

Right side: $\frac{4(2)}{3(2)} = \frac{8}{6} = \frac{4}{3}$.

Since $2 \neq \frac{4}{3}$, the statement is false.

The statement confuses exponents (repeated multiplication) with multiplication by a constant (repeated addition).

Conclusion:

The statement $\frac{a^4}{b^3} = \frac{a + a + a + a}{b + b + b}$ is False.

Given:

The statement: $a^b > b^a$ is true if $a = 3$ and $b = 4$, and false if $a = 2$ and $b = 3$.

To State:

Whether the given statement is True or False.

Solution:

We need to evaluate the inequality $a^b > b^a$ for the two given pairs of values for $a$ and $b$ and check if the results match the claims made in the statement.

Case 1: $a = 3$ and $b = 4$

We need to check if $a^b > b^a$ is true, i.e., if $3^4 > 4^3$.

Calculate $3^4$:

$3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81$

Calculate $4^3$:

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$

Now compare the values:

$81 > 64$

This inequality is True. The statement claims that $a^b > b^a$ is true when $a=3$ and $b=4$. This part of the statement is correct.

Case 2: $a = 2$ and $b = 3$

We need to check if $a^b > b^a$ is false, i.e., if $2^3 > 3^2$ is false.

Calculate $2^3$:

$2^3 = 2 \times 2 \times 2 = 4 \times 2 = 8$

Calculate $3^2$:

$3^2 = 3 \times 3 = 9$

Now compare the values:

$8 > 9$

This inequality is False.

The statement claims that $a^b > b^a$ is false when $a=2$ and $b=3$. Since the inequality $8 > 9$ is indeed false, this part of the statement is also correct.

The given statement is a compound statement that asserts two conditions: 1) the inequality holds for $(a,b) = (3,4)$ and 2) the inequality does not hold for $(a,b) = (2,3)$. We found that both of these conditions are true.

Conclusion:

Since the inequality $a^b > b^a$ is indeed true for $a=3, b=4$ and false for $a=2, b=3$, the complete statement is True.

Given:

The initial number is $3^3$.

The desired product is $3^7$.

To Find:

The number by which $3^3$ must be multiplied to obtain $3^7$.

Solution:

Let the unknown number be $N$.

According to the problem statement, when we multiply $3^3$ by $N$, the product is $3^7$. We can write this as an equation:

To find the value of $N$, we need to isolate $N$ in the equation. We can do this by dividing both sides of the equation by $3^3$ (since $3^3 \neq 0$).

Now, we can simplify the right side of the equation using the rule of exponents for division with the same base. The rule states that for any non-zero base $a$ and integers $m$ and $n$, $\frac{a^m}{a^n} = a^{m-n}$.

In this case, the base is $a=3$, the exponent in the numerator is $m=7$, and the exponent in the denominator is $n=3$.

Subtract the exponents:

$7 - 3 = 4$

So, the value of $N$ is:

$N = 3^4$

To find the actual numerical value, we calculate $3^4$:

$3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81$

So, the number is 81.

We can check this: $3^3 \times 3^4 = 27 \times 81$. This multiplication is tedious, but using exponents, $3^3 \times 3^4 = 3^{3+4} = 3^7$, which is the desired product.

Answer:

We should multiply $3^3$ by $3^4$ (or 81) to get $3^7$.

Given:

The equation $\left( \frac{5}{3} \right)^5 \times \left( \frac{5}{3} \right)^{11} = \left( \frac{5}{3} \right)^{8x}$.

To Find:

The value of $x$ that satisfies the given equation.

Solution:

We are given the equation $\left( \frac{5}{3} \right)^5 \times \left( \frac{5}{3} \right)^{11} = \left( \frac{5}{3} \right)^{8x}$.

First, we simplify the left side of the equation using the rule of exponents for multiplication with the same base: $a^m \times a^n = a^{m+n}$.

In the expression $\left( \frac{5}{3} \right)^5 \times \left( \frac{5}{3} \right)^{11}$, the base is $a = \frac{5}{3}$, the exponents are $m = 5$ and $n = 11$.

Applying the rule to the left side:

Adding the exponents:

$5 + 11 = 16$

So, the left side simplifies to $\left( \frac{5}{3} \right)^{16}$.

Now, the equation becomes:

$\left( \frac{5}{3} \right)^{16} = \left( \frac{5}{3} \right)^{8x}$

We have an equation where the bases are the same and are not equal to 0 or 1. For such an equation to hold true, the exponents must be equal.

Therefore, we can equate the exponents:

Now, we solve this linear equation for $x$. To isolate $x$, we divide both sides of the equation by 8:

$x = \frac{16}{8}$

$x = 2$

Thus, the value of $x$ is 2.

Answer:

The value of $x$ is 2.

Given:

The number 648.

To Find:

Express 648 in exponential notation.

Solution:

To express a number in exponential notation, we need to find its prime factorization. This means writing the number as a product of its prime factors, each raised to a power.

We will find the prime factors of 648 by dividing it successively by prime numbers starting from the smallest prime, 2.

Perform the prime factorization:

$\begin{array}{c|cc} 2 & 648 \\ \hline 2 & 324 \\ \hline 2 & 162 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$

From the prime factorization, we can see that:

$648 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$

Now, we write this product using exponents:

The factor 2 appears 3 times, so we write $2^3$.

The factor 3 appears 4 times, so we write $3^4$.

Thus, the exponential notation for 648 is the product of these exponential forms:

$648 = 2^3 \times 3^4$

Answer:

The exponential notation for 648 is $2^3 \times 3^4$.

Given:

The number 2,36,00,000.

To Find:

Express the given number in standard form.

Solution:

Standard form (also known as scientific notation) is a way of writing numbers as a product of a number between 1 and 10 (inclusive of 1 but not 10) and a power of 10.

A number in standard form is written as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.

The given number is 2,36,00,000.

In this number, the decimal point is implicitly at the end: 23600000.

To write it in standard form, we need to move the decimal point so that there is only one non-zero digit to the left of the decimal point. The first non-zero digit from the left is 2.

So, we need to move the decimal point from its current position to a position after the digit 2.

The original position of the decimal is after the last zero. We move it to the left past each digit until it is after 2:

$23600000.$

Moving past 0 (1st place)

Moving past 0 (2nd place)

Moving past 0 (3rd place)

Moving past 0 (4th place)

Moving past 0 (5th place)

Moving past 6 (6th place)

Moving past 3 (7th place)

The new position of the decimal point is after 2. The number becomes 2.3600000, which is written as 2.36 (since trailing zeros after the decimal point are not necessary unless they are significant).

The decimal point was moved 7 places to the left.

When the decimal point is moved $n$ places to the left, the number is multiplied by $10^n$. In this case, $n=7$.

So, $23600000 = 2.36 \times 10^7$.

Here, $a = 2.36$, which satisfies $1 \leq 2.36 < 10$, and $n = 7$, which is an integer.

Thus, the standard form of 2,36,00,000 is $2.36 \times 10^7$.

Answer:

The standard form of 2,36,00,000 is $2.36 \times 10^7$.

Given:

The two numbers to compare are $3^{12}$ and $6^6$.

To Find:

Determine which of the two numbers, $3^{12}$ or $6^6$, is larger.

Solution:

To compare $3^{12}$ and $6^6$, we can try to express them with the same exponent or the same base.

Let's look at the exponents, 12 and 6. We notice that $12 = 2 \times 6$. This suggests we can rewrite $3^{12}$ using an exponent of 6.

We use the property of exponents $(a^m)^n = a^{m \times n}$. Reversing this, $a^{m \times n} = (a^m)^n$.

So, $3^{12} = 3^{2 \times 6}$.

Using the property, we can write this as:

Now, we calculate the value inside the parenthesis:

$3^2 = 3 \times 3 = 9$

So, $3^{12} = 9^6$.

Now the problem is reduced to comparing $9^6$ and $6^6$.

We are comparing two numbers with the same exponent (6). When comparing powers with the same positive integer exponent, the power with the larger base is the larger number.

We compare the bases: 9 and 6.

Since $9 > 6$, it follows that $9^6 > 6^6$.

Since $3^{12} = 9^6$, we can conclude that $3^{12} > 6^6$.

Alternatively, we could calculate the values:

$3^{12} = 531441$ (calculated in thinking process)

$6^6 = 46656$ (calculated in thinking process)

Comparing the values: $531441 > 46656$. This confirms that $3^{12}$ is larger.

Conclusion:

Comparing $3^{12} = (3^2)^6 = 9^6$ with $6^6$, since the exponents are equal and the base $9$ is greater than the base $6$, we have $9^6 > 6^6$. Therefore, $3^{12}$ is larger than $6^6$.

The larger number is $3^{12}$.

Given:

The equation $\left( \frac{1}{5} \right)^5 \times \left( \frac{1}{5} \right)^{19} = \left( \frac{1}{5} \right)^{8x}$.

To Find:

The value of $x$ that satisfies the given equation.

Solution:

We are given the equation $\left( \frac{1}{5} \right)^5 \times \left( \frac{1}{5} \right)^{19} = \left( \frac{1}{5} \right)^{8x}$.

First, we simplify the left side of the equation using the rule of exponents for multiplication with the same base: $a^m \times a^n = a^{m+n}$.

In the expression $\left( \frac{1}{5} \right)^5 \times \left( \frac{1}{5} \right)^{19}$, the base is $a = \frac{1}{5}$, the exponents are $m = 5$ and $n = 19$.

Applying the rule to the left side:

Adding the exponents:

$5 + 19 = 24$

So, the left side simplifies to $\left( \frac{1}{5} \right)^{24}$.

Now, the equation becomes:

$\left( \frac{1}{5} \right)^{24} = \left( \frac{1}{5} \right)^{8x}$

We have an equation where the bases are the same $\left(\frac{1}{5}\right)$ and are not equal to 0 or 1. For such an equation to be true, the exponents must be equal.

Therefore, we can equate the exponents:

Now, we solve this linear equation for $x$. To isolate $x$, we divide both sides of the equation by 8:

$x = \frac{24}{8}$

$x = 3$

Thus, the value of $x$ is 3.

Answer:

The value of $x$ is 3.

The given expression is $[(-3)^2]^3$.

We use the law of exponents which states that for any non-zero integer $a$ and positive integers $m$ and $n$, $(a^m)^n = a^{m \times n}$.

Applying this rule to the given expression, we have:

$[(-3)^2]^3 = (-3)^{2 \times 3}$

$ = (-3)^6$

Thus, $[(-3)^2]^3$ is equal to $(-3)^6$.

Comparing this with the given options:

(a) $(-3)^8$

(b) $(-3)^6$

(c) $(-3)^5$

(d) $(-3)^{23}$

The correct option is (b).

The given expression is $x^8 \div x^2$. Here, $x$ is a non-zero rational number.

We use the law of exponents which states that for any non-zero rational number $a$ and integers $m$ and $n$, $a^m \div a^n = a^{m-n}$.

Applying this rule to the given expression, with $a=x$, $m=8$, and $n=2$, we have:

$x^8 \div x^2 = x^{8-2}$

$ = x^6$

Thus, $x^8 \div x^2$ is equal to $x^6$.

Comparing this with the given options:

(a) $x^4$

(b) $x^6$

(c) $x^{10}$

(d) $x^{16}$

The correct option is (b).

Let $x$ be a non-zero rational number.

The square of $x$ is $x^2$.

The cube of $x$ is $x^3$.

The product of the square of $x$ and the cube of $x$ is $x^2 \times x^3$.

We use the law of exponents which states that for any non-zero rational number $a$ and integers $m$ and $n$, $a^m \times a^n = a^{m+n}$.

Applying this rule to the product $x^2 \times x^3$, with $a=x$, $m=2$, and $n=3$, we have:

$x^2 \times x^3 = x^{2+3}$

$= x^5$

The result is $x^5$, which represents the fifth power of $x$.

Comparing this with the given options:

(a) second power of x ($x^2$)

(b) third power of x ($x^3$)

(c) fifth power of x ($x^5$)

(d) sixth power of x ($x^6$)

The correct option is (c).

The given expression is $x^5 \div y^5$. Here, $x$ and $y$ are non-zero rational numbers.

The expression can be written as $\frac{x^5}{y^5}$.

We use the law of exponents which states that for any non-zero rational numbers $a$ and $b$, and any integer $m$, $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.

Applying this rule to the given expression, with $a=x$, $b=y$, and $m=5$, we have:

$\frac{x^5}{y^5} = \left(\frac{x}{y}\right)^5$

This is equivalent to $(x \div y)^5$.

Thus, $x^5 \div y^5$ is equal to $(x \div y)^5$.

Comparing this with the given options:

(a) $(x \div y)^1$

(b) $(x \div y)^0$

(c) $(x \div y)^5$

(d) $(x \div y)^{10}$

The correct option is (c).

The given expression is $a^m \times a^n$.

This is a fundamental property or law of exponents.

The law states that when multiplying exponential expressions with the same base, we add the exponents.

Symbolically, for any base $a$ and exponents $m$ and $n$, the rule is:

$a^m \times a^n = a^{m+n}$

Comparing this result with the given options:

(a) $(a^2)^{mn} = a^{2mn}$ (using $(a^x)^y = a^{xy}$)

(b) $a^{m-n}$

(c) $a^{m+n}$

(d) $a^{mn}$

The correct option that matches our simplified expression is (c).

The given expression is $(1^0 + 2^0 + 3^0)$.

We use the property of exponents which states that any non-zero number raised to the power of 0 is equal to 1.

Mathematically, for any $a \neq 0$, $a^0 = 1$.

Applying this property to each term in the expression:

$1^0 = 1$ (since $1 \neq 0$)

$2^0 = 1$ (since $2 \neq 0$)

$3^0 = 1$ (since $3 \neq 0$)

Now, substitute these values back into the expression:

$(1^0 + 2^0 + 3^0) = (1 + 1 + 1)$

$ = 3$

Thus, $(1^0 + 2^0 + 3^0)$ is equal to 3.

Comparing this with the given options:

(a) 0

(b) 1

(c) 3

(d) 6

The correct option is (c).

The given expression is $\frac{10^{22} + 10^{20}}{10^{20}}$.

We can rewrite the numerator by factoring out the common term $10^{20}$.

Note that $10^{22} = 10^{20 + 2} = 10^{20} \times 10^2$.

So, the numerator $10^{22} + 10^{20}$ can be written as:

$10^{20} \times 10^2 + 10^{20} \times 1$

Factoring out $10^{20}$, we get:

$10^{20}(10^2 + 1)$

Now, substitute this back into the original expression:

$\frac{10^{20}(10^2 + 1)}{10^{20}}$

We can cancel out the common factor $10^{20}$ from the numerator and the denominator, since $10^{20} \neq 0$.

The expression simplifies to:

$10^2 + 1$

Calculate the value of $10^2$:

$10^2 = 10 \times 10 = 100$

Now, substitute this value back:

$100 + 1 = 101$

Thus, the value of the expression is 101.

Comparing this with the given options:

(a) 10

(b) $10^{42}$

(c) 101

(d) $10^{22}$

The correct option is (c).

The given number is 12345.

The standard form of a number is expressed as a number between 1 and 10 (inclusive of 1) multiplied by a power of 10.

That is, a number is written in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

To convert 12345 into standard form, we need to place the decimal point such that the resulting number is between 1 and 10.

The number 12345 has an implied decimal point after the last digit: $12345.0$.

To get a number between 1 and 10, we move the decimal point to the left until it is after the first digit (1).

Moving the decimal point from its original position to the new position (between 1 and 2) involves moving it 4 places to the left.

$12345. \implies 1.2345$

Since we moved the decimal point 4 places to the left, we multiply the resulting number by $10^4$.

So, $12345 = 1.2345 \times 10^4$.

Thus, the standard form of 12345 is $1.2345 \times 10^4$.

Comparing this with the given options:

(a) $1234.5 \times 10^1 = 12345$

(b) $123.45 \times 10^2 = 12345$

(c) $12.345 \times 10^3 = 12345$

(d) $1.2345 \times 10^4 = 12345$

All options represent the correct value, but only option (d) is in the standard form where the first part of the number is between 1 and 10.

The correct option is (d).

The given equation is:

We need to find the value of $K$. Let's simplify the left-hand side (LHS) of the equation.

The LHS is $2^{1998} – 2^{1997} – 2^{1996} + 2^{1995}$.

We can express each term in relation to the lowest power of 2 present, which is $2^{1995}$.

Using the property $a^{m+n} = a^m \cdot a^n$, we have:

$2^{1998} = 2^{1995 + 3} = 2^{1995} \cdot 2^3$

$2^{1997} = 2^{1995 + 2} = 2^{1995} \cdot 2^2$

$2^{1996} = 2^{1995 + 1} = 2^{1995} \cdot 2^1$

$2^{1995} = 2^{1995} \cdot 1$

Substitute these expressions back into the LHS:

$LHS = (2^{1995} \cdot 2^3) - (2^{1995} \cdot 2^2) - (2^{1995} \cdot 2^1) + (2^{1995} \cdot 1)$

Factor out the common term $2^{1995}$:

$LHS = 2^{1995} (2^3 - 2^2 - 2^1 + 1)$

Now, evaluate the terms inside the parenthesis:

$2^3 = 8$

$2^2 = 4$

$2^1 = 2$

The expression inside the parenthesis becomes:

$8 - 4 - 2 + 1 = (8 - 4) - 2 + 1 = 4 - 2 + 1 = 2 + 1 = 3$

So, the simplified LHS is $2^{1995} \cdot 3$.

Now, equate the simplified LHS with the RHS of equation (i):

$2^{1995} \cdot 3 = K \cdot 2^{1995}$

To find $K$, we can divide both sides of the equation by $2^{1995}$ (since $2^{1995} \neq 0$).

$\frac{2^{1995} \cdot 3}{2^{1995}} = \frac{K \cdot 2^{1995}}{2^{1995}}$

$3 = K$

Thus, the value of $K$ is 3.

Comparing this with the given options:

(a) 1

(b) 2

(c) 3

(d) 4

The correct option is (c).

We need to evaluate each given expression to determine which one is equal to 1.

Recall the property of exponents that for any non-zero number $a$, $a^0 = 1$.

Let's evaluate each option:

(a) $2^0 + 3^0 + 4^0$

Using the property, $2^0 = 1$, $3^0 = 1$, and $4^0 = 1$.

$2^0 + 3^0 + 4^0 = 1 + 1 + 1 = 3$

This is not equal to 1.

(b) $2^0 \times 3^0 \times 4^0$

Using the property, $2^0 = 1$, $3^0 = 1$, and $4^0 = 1$.

$2^0 \times 3^0 \times 4^0 = 1 \times 1 \times 1 = 1$

This is equal to 1.

(c) $(3^0 – 2^0) \times 4^0$

Using the property, $3^0 = 1$, $2^0 = 1$, and $4^0 = 1$.

$(3^0 – 2^0) \times 4^0 = (1 - 1) \times 1 = 0 \times 1 = 0$

This is not equal to 1.

(d) $(3^0 – 2^0) \times (3^0 + 2^0)$

Using the property, $3^0 = 1$ and $2^0 = 1$.

$(3^0 – 2^0) \times (3^0 + 2^0) = (1 - 1) \times (1 + 1) = 0 \times 2 = 0$

This is not equal to 1.

Comparing the results, only option (b) evaluates to 1.

The correct option is (b).

The given number is 72105.4.

Its standard form is given as $7.21054 \times 10^n$.

The standard form of a number is written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

In the given standard form $7.21054 \times 10^n$, the value $a$ is 7.21054, which satisfies the condition $1 \leq 7.21054 < 10$.

To convert 72105.4 into the form $7.21054 \times 10^n$, we need to determine how many places and in which direction the decimal point was moved from the original number 72105.4 to get 7.21054.

The original number is 72105.4.

The number in standard form is 7.21054.

To get from 72105.4 to 7.21054, the decimal point has been moved 4 places to the left (past the digits 5, 0, 1, and 2).

When the decimal point is moved $n$ places to the left, the number is multiplied by $10^n$.

In this case, the decimal point is moved 4 places to the left, so the power of 10 is $10^4$.

Therefore, $72105.4 = 7.21054 \times 10^4$.

Comparing this with the given standard form $7.21054 \times 10^n$, we find that $n = 4$.

Thus, the value of $n$ is 4.

Comparing this with the given options:

(a) 2

(b) 3

(c) 4

(d) 5

The correct option is (c).

We need to find the square of $\left( \frac{-2}{3} \right)$.

The square of a number means raising the number to the power of 2.

So, we need to calculate $\left( \frac{-2}{3} \right)^2$.

Using the property of exponents $(a/b)^m = a^m / b^m$, we can write:

$\left( \frac{-2}{3} \right)^2 = \frac{(-2)^2}{3^2}$

Now, we calculate the values of the numerator and the denominator:

$(-2)^2 = (-2) \times (-2) = 4$

$3^2 = 3 \times 3 = 9$

Substitute these values back into the expression:

$\frac{(-2)^2}{3^2} = \frac{4}{9}$

Thus, the square of $\left( \frac{-2}{3} \right)$ is $\frac{4}{9}$.

Comparing this with the given options:

(a) $\frac{-2}{3}$

(b) $\frac{2}{3}$

(c) $\frac{-4}{9}$

(d) $\frac{4}{9}$

The correct option is (d).

We need to find the cube of $\left( \frac{-1}{4} \right)$.

The cube of a number means raising the number to the power of 3.

So, we need to calculate $\left( \frac{-1}{4} \right)^3$.

Using the property of exponents $(a/b)^m = a^m / b^m$, we can write:

$\left( \frac{-1}{4} \right)^3 = \frac{(-1)^3}{4^3}$

Now, we calculate the values of the numerator and the denominator:

$(-1)^3 = (-1) \times (-1) \times (-1) = 1 \times (-1) = -1$

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$

Substitute these values back into the expression:

$\frac{(-1)^3}{4^3} = \frac{-1}{64}$

Thus, the cube of $\left( \frac{-1}{4} \right)$ is $\frac{-1}{64}$.

Comparing this with the given options:

(a) $\frac{-1}{12}$

(b) $\frac{1}{16}$

(c) $\frac{-1}{64}$

(d) $\frac{1}{64}$

The correct option is (c).

The given expression is $\left( \frac{-5}{4} \right)^4$.

We need to determine which of the given options is not equal to this expression.

Let's evaluate the given expression first.

$\left( \frac{-5}{4} \right)^4 = \frac{(-5)^4}{4^4}$

Since the exponent is an even number (4), the result of raising a negative number to this power is positive.

$(-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 25 \times 25 = 625$

$4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256$

So, $\left( \frac{-5}{4} \right)^4 = \frac{625}{256}$.

Now, let's evaluate each option:

(a) $\frac{(−5)^4}{4^4}$

As calculated above, $(-5)^4 = 625$ and $4^4 = 256$.

So, $\frac{(−5)^4}{4^4} = \frac{625}{256}$. This is equal to the original expression.

(b) $\frac{5^4}{(-4)^4}$

$5^4 = 5 \times 5 \times 5 \times 5 = 625$

$(-4)^4 = (-4) \times (-4) \times (-4) \times (-4) = 16 \times 16 = 256$. Note that $(-4)^4 = 4^4$.

So, $\frac{5^4}{(-4)^4} = \frac{625}{256}$. This is equal to the original expression.

(c) $-\frac{5^4}{4^4}$

$5^4 = 625$ and $4^4 = 256$.

So, $-\frac{5^4}{4^4} = -\frac{625}{256}$.

This is a negative value, while the original expression $\left( \frac{-5}{4} \right)^4 = \frac{625}{256}$ is a positive value.

Therefore, this option is not equal to the original expression.

(d) $\left( -\frac{5}{4} \right) × \left( -\frac{5}{4} \right) × \left( -\frac{5}{4} \right) × \left( -\frac{5}{4} \right)$

This is the expanded form of $\left( \frac{-5}{4} \right)^4$. By the definition of exponentiation, this product is equal to $\left( \frac{-5}{4} \right)^4$.

Multiplying four negative numbers results in a positive number.

Numerator product: $(-5) \times (-5) \times (-5) \times (-5) = 625$

Denominator product: $4 \times 4 \times 4 \times 4 = 256$

The product is $\frac{625}{256}$. This is equal to the original expression.

Comparing the values, options (a), (b), and (d) are equal to $\frac{625}{256}$, while option (c) is equal to $-\frac{625}{256}$.

Thus, the expression which is not equal to $\left( \frac{-5}{4} \right)^4$ is $-\frac{5^4}{4^4}$.

The correct option is (c).

We need to evaluate each expression to find which one is not equal to 1.

(a) $\frac{2^{3} \;×\; 3^{2}}{4 \;×\; 18}$

Calculate the terms in the numerator and denominator:

$2^3 = 2 \times 2 \times 2 = 8$

$3^2 = 3 \times 3 = 9$

$4 \times 18 = 72$

The expression becomes: $\frac{8 \times 9}{72} = \frac{72}{72} = 1$.

Alternatively, using prime factorisation:

$\frac{2^3 \times 3^2}{2^2 \times (2 \times 3^2)} = \frac{2^3 \times 3^2}{2^3 \times 3^2}$

Using the property $a^m/a^n = a^{m-n}$ and $a^m b^n / a^m b^n = 1$:

$\frac{\cancel{2^3} \times \cancel{3^2}}{\cancel{2^3} \times \cancel{3^2}} = 1$.

So, option (a) is equal to 1.

(b) $[(-2)^3 \times (-2)^4] \div (-2)^7$

Using the property $a^m \times a^n = a^{m+n}$ for the terms inside the brackets:

$(-2)^3 \times (-2)^4 = (-2)^{3+4} = (-2)^7$

The expression becomes: $(-2)^7 \div (-2)^7$

Using the property $a^m \div a^m = a^{m-m} = a^0 = 1$ (for $a \neq 0$):

$(-2)^7 \div (-2)^7 = (-2)^{7-7} = (-2)^0 = 1$.

So, option (b) is equal to 1.

(c) $\frac{3^{0} \;×\; 5^{3}}{5 \;×\; 25}$

Using the property $a^0 = 1$ for $a \neq 0$, $3^0 = 1$.

$5^3 = 5 \times 5 \times 5 = 125$

$5 \times 25 = 125$

The expression becomes: $\frac{1 \times 125}{125} = \frac{125}{125} = 1$.

Alternatively, using exponents:

$\frac{3^0 \times 5^3}{5^1 \times 5^2} = \frac{1 \times 5^3}{5^{1+2}} = \frac{5^3}{5^3}$

Using the property $a^m/a^m = 1$:

$\frac{\cancel{5^3}}{\cancel{5^3}} = 1$.

So, option (c) is equal to 1.

(d) $\frac{2^{4}}{(7^{0} \;+\; 3^{0})^3}$

Using the property $a^0 = 1$ for $a \neq 0$:

$7^0 = 1$

$3^0 = 1$

The denominator base is $(7^0 + 3^0) = (1 + 1) = 2$.

The denominator is $(1+1)^3 = 2^3 = 2 \times 2 \times 2 = 8$.

The numerator is $2^4 = 2 \times 2 \times 2 \times 2 = 16$.

The expression becomes: $\frac{16}{8} = 2$.

So, option (d) is equal to 2.

Comparing the results, options (a), (b), and (c) are equal to 1, while option (d) is equal to 2.

Thus, the expression which is not equal to 1 is given in option (d).

The correct option is (d).

The given expression is $\left( \frac{2}{3} \right)^{3} \times \left( \frac{5}{7} \right)^{3}$.

We use the law of exponents which states that for any rational numbers $a$ and $b$, and any integer $m$, $a^m \times b^m = (a \times b)^m$.

In this case, $a = \frac{2}{3}$, $b = \frac{5}{7}$, and $m = 3$.

Applying this rule, we get:

$\left( \frac{2}{3} \right)^{3} \times \left( \frac{5}{7} \right)^{3} = \left( \frac{2}{3} \times \frac{5}{7} \right)^{3}$

Thus, the expression is equal to $\left( \frac{2}{3} \times \frac{5}{7} \right)^{3}$.

Comparing this with the given options:

(a) $\left( \frac{2}{3} \times \frac{5}{7} \right)^{9}$

(b) $\left( \frac{2}{3} \times \frac{5}{7} \right)^{6}$

(c) $\left( \frac{2}{3} \times \frac{5}{7} \right)^{3}$

(d) $\left( \frac{2}{3} \times \frac{5}{7} \right)^{0}$

The correct option is (c).

The given number is 829030000.

Its standard form is given as $K \times 10^8$.

The standard form of a number is expressed as a number between 1 and 10 (inclusive of 1) multiplied by a power of 10.

That is, a number is written in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

In the given form $K \times 10^8$, the power of 10 is 8, so $n=8$. This means the number $K$ should be between 1 and 10.

To convert 829030000 into standard form, we need to place the decimal point such that the resulting number ($K$) is between 1 and 10.

The number 829030000 has an implied decimal point after the last digit: $829030000.0$.

To get a number between 1 and 10, we move the decimal point to the left until it is after the first non-zero digit (8).

$829030000. \implies 8.29030000$

The number of places the decimal point was moved to the left is 8 (past the digits 0, 0, 0, 0, 3, 0, 9, and 2).

Since the decimal point was moved 8 places to the left, the number in standard form is $8.2903 \times 10^8$ (trailing zeros after the decimal point can be omitted).

Comparing this with the given form $K \times 10^8$, we have:

$K = 8.2903$

Let's verify if $K$ is between 1 and 10: $1 \leq 8.2903 < 10$. This is true.

Thus, the value of $K$ is 8.2903.

Comparing this with the given options:

(a) 82903 (Not between 1 and 10)

(b) 829.03 (Not between 1 and 10)

(c) 82.903 (Not between 1 and 10)

(d) 8.2903 (Between 1 and 10)

The correct option is (d).

We need to find which of the given options has the largest value. Let's evaluate each option.

(a) 0.0001

This value is already in decimal form.

In scientific notation, $0.0001 = 1 \times 10^{-4}$.

In fractional form, $0.0001 = \frac{1}{10000}$.

(b) $\frac{1}{1000}$

This value is in fractional form.

In decimal form, $\frac{1}{1000} = 0.001$.

In scientific notation, $\frac{1}{1000} = \frac{1}{10^3} = 10^{-3} = 1 \times 10^{-3}$.

(c) $\frac{1}{10^6}$

This value is in exponential fractional form.

In decimal form, $\frac{1}{10^6} = \frac{1}{1000000} = 0.000001$.

In scientific notation, $\frac{1}{10^6} = 10^{-6} = 1 \times 10^{-6}$.

(d) $\frac{1}{10^6}$ ÷ 0.1

First, evaluate the division. Recall that dividing by a number is equivalent to multiplying by its reciprocal.

$0.1 = \frac{1}{10}$

So, $\frac{1}{10^6} \div 0.1 = \frac{1}{10^6} \div \frac{1}{10}$

$= \frac{1}{10^6} \times \frac{10}{1}$

$= \frac{10}{10^6}$

Using the property $a^m/a^n = a^{m-n}$:

$= 10^{1-6}$

$= 10^{-5}$

In decimal form, $10^{-5} = 0.00001$.

In fractional form, $10^{-5} = \frac{1}{10^5} = \frac{1}{100000}$.

Now let's compare the values of all options, preferably in decimal form:

(a) 0.0001

(b) 0.001

(c) 0.000001

(d) 0.00001

Comparing these decimal values:

$0.001 > 0.0001 > 0.00001 > 0.000001$

The largest value is 0.001.

Alternatively, comparing in scientific notation:

(a) $1 \times 10^{-4}$

(b) $1 \times 10^{-3}$

(c) $1 \times 10^{-6}$

(d) $1 \times 10^{-5}$

Comparing powers of 10 with negative exponents, the largest value corresponds to the least negative exponent. The exponents are -4, -3, -6, -5.

The largest exponent is -3.

So, $10^{-3}$ is the largest value.

The value $0.001$ corresponds to option (b).

The correct option is (b).

The number is 72 crore.

In the Indian numbering system, 1 crore is equal to 10,000,000 (ten million).

So, 72 crore means 72 times 1 crore.

72 crore = $72 \times 10,000,000$

= $72,000,000$

The standard form of a number is written as $a \times 10^n$, where $a$ is a number such that $1 \leq a < 10$, and $n$ is an integer.

To convert 72,000,000 into standard form, we need to place the decimal point after the first non-zero digit (which is 7).

The number 72,000,000 can be written with an implied decimal point at the end: $72000000.$

We move the decimal point to the left until it is between 7 and 2:

$72000000. \implies 7.2000000$

The decimal point was moved 7 places to the left (past the seven 0s and the digit 2).

The number of places the decimal is moved to the left gives the positive power of 10.

So, the power of 10 is $10^7$.

Therefore, $72,000,000 = 7.2 \times 10^7$.

This is in standard form because $1 \leq 7.2 < 10$ and the power of 10 is an integer.

Comparing this with the given options:

(a) $72 \times 10^7$ (Value is correct, but 72 is not between 1 and 10, so not standard form)

(b) $72 \times 10^8$ (Incorrect value: $72 \times 100,000,000 = 7,200,000,000$)

(c) $7.2 \times 10^8$ (Incorrect value: $7.2 \times 100,000,000 = 720,000,000$)

(d) $7.2 \times 10^7$ (Value is correct: $7.2 \times 10,000,000 = 72,000,000$, and it is in standard form)

The correct option is (d).

The given expression is $\left( \frac{a}{b} \right)^{m} \div \left( \frac{a}{b} \right)^{n}$, where $a$ and $b$ are non-zero numbers and $m > n$.

We use the law of exponents which states that for any non-zero base $x$ and integers $m$ and $n$, $x^m \div x^n = x^{m-n}$.

In this expression, the base is $x = \frac{a}{b}$, and the exponents are $m$ and $n$.

Applying the rule, we get:

$\left( \frac{a}{b} \right)^{m} \div \left( \frac{a}{b} \right)^{n} = \left( \frac{a}{b} \right)^{m-n}$

The condition $m > n$ ensures that the resulting exponent $m-n$ is a positive integer, which is consistent with the usual definition of division of powers.

Thus, the expression is equal to $\left( \frac{a}{b} \right)^{m-n}$.

Comparing this with the given options:

(a) $\left( \frac{a}{b} \right)^{mn}$

(b) $\left( \frac{a}{b} \right)^{m+n}$

(c) $\left( \frac{a}{b} \right)^{m-n}$

(d) $\left( \left( \frac{a}{b} \right)^{m} \right)^n = \left( \frac{a}{b} \right)^{mn}$ (using the property $(x^p)^q = x^{pq}$)

The correct option that matches our simplified expression is (c).

We need to evaluate each statement to determine which one is not true.

(a) $3^2 > 2^3$

Calculate the values:

$3^2 = 3 \times 3 = 9$

$2^3 = 2 \times 2 \times 2 = 8$

Compare the values: $9 > 8$. This statement is true.

(b) $4^3 = 2^6$

Calculate the values:

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$

$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 8 \times 8 = 64$

Alternatively, recognize that $4 = 2^2$. So, $4^3 = (2^2)^3$. Using the property $(a^m)^n = a^{mn}$, we get $(2^2)^3 = 2^{2 \times 3} = 2^6$.

Compare the values: $64 = 64$. This statement is true.

(c) $3^3 = 9$

Calculate the value:

$3^3 = 3 \times 3 \times 3 = 9 \times 3 = 27$

Compare the values: $27 = 9$. This statement is false, as $27 \neq 9$.

(d) $2^5 > 5^2$

Calculate the values:

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 4 \times 4 \times 2 = 16 \times 2 = 32$

$5^2 = 5 \times 5 = 25$

Compare the values: $32 > 25$. This statement is true.

Based on the evaluations, the statement that is not true is $3^3 = 9$.

The correct option is (c).

To Find: The power of 8 that is equal to $2^6$.

Solution:

Let the required power of 8 be $p$. We want to find the value of $p$ such that:

We need to express both sides of the equation with the same base.

The base on the right-hand side (RHS) is 2.

We can express the base on the left-hand side (LHS), which is 8, as a power of 2.

$8 = 2 \times 2 \times 2 = 2^3$

Substitute $8 = 2^3$ into equation (i):

$(2^3)^p = 2^6$

Using the law of exponents which states that for any base $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$, we apply this to the LHS:

$(2^3)^p = 2^{3 \times p} = 2^{3p}$

Now, substitute this back into the equation:

$2^{3p} = 2^6$

Since the bases are equal, the exponents must be equal for the equality to hold:

$3p = 6$

Solve for $p$ by dividing both sides by 3:

$p = \frac{6}{3}$

$p = 2$

So, the 2nd power of 8 is equal to $2^6$.

Let's check the values:

$8^2 = 8 \times 8 = 64$

$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$

Thus, the power of 8 that is equal to $2^6$ is 2.

Comparing this with the given options:

(a) 3

(b) 2

(c) 1

(d) 4

The correct option is (b).

The given equation is $(-2)^{31} \times (-2)^{13} = (-2)^{\square}$.

We use the law of exponents for multiplication with the same base, which states that for any non-zero number $a$ and integers $m$ and $n$, $a^m \times a^n = a^{m+n}$.

In this equation, the base is $a = -2$, and the exponents are $m = 31$ and $n = 13$.

Applying the rule to the left-hand side (LHS):

$(-2)^{31} \times (-2)^{13} = (-2)^{31 + 13}$

Now, perform the addition of the exponents:

$31 + 13 = 44$

So, the LHS simplifies to $(-2)^{44}$.

Equating the simplified LHS to the right-hand side (RHS):

$(-2)^{44} = (-2)^{\square}$

Since the bases are the same ($ -2 $), the exponents must be equal.

Therefore, the missing exponent is 44.

The filled-in statement is $(–2)^{31} \times (–2)^{13} = (−2)^{44}$.

The blank should be filled with 44.

The given equation is $(-3)^8 \div (-3)^5 = (-3)^{\square}$.

We use the law of exponents for division with the same base, which states that for any non-zero number $a$ and integers $m$ and $n$, $a^m \div a^n = a^{m-n}$.

In this equation, the base is $a = -3$, and the exponents are $m = 8$ and $n = 5$.

Applying the rule to the left-hand side (LHS):

$(-3)^8 \div (-3)^5 = (-3)^{8 - 5}$

Now, perform the subtraction of the exponents:

$8 - 5 = 3$

So, the LHS simplifies to $(-3)^3$.

Equating the simplified LHS to the right-hand side (RHS):

$(-3)^3 = (-3)^{\square}$

Since the bases are the same ($ -3 $), the exponents must be equal.

Therefore, the missing exponent is 3.

The filled-in statement is $(–3)^{8} \div (–3)^{5} = (–3)^{3}$.

The blank should be filled with 3.

The given equation is $\left( \frac{11}{15} \right)^{4} \times (\square)^{5} = \left( \frac{11}{15} \right)^{9}$.

Let the missing term in the parenthesis be $x$. The equation is $\left( \frac{11}{15} \right)^{4} \times (x)^{5} = \left( \frac{11}{15} \right)^{9}$.

We observe that the base on the LHS and RHS involves $\left( \frac{11}{15} \right)$. It is likely that the missing base is also $\frac{11}{15}$.

Assume the missing base is $\frac{11}{15}$. The equation becomes $\left( \frac{11}{15} \right)^{4} \times \left( \frac{11}{15} \right)^{5} = \left( \frac{11}{15} \right)^{9}$.

We use the law of exponents for multiplication with the same base, which states that for any non-zero number $a$ and integers $m$ and $n$, $a^m \times a^n = a^{m+n}$.

Apply this rule to the LHS of the assumed equation:

$\left( \frac{11}{15} \right)^{4} \times \left( \frac{11}{15} \right)^{5} = \left( \frac{11}{15} \right)^{4+5}$

$= \left( \frac{11}{15} \right)^{9}$

The simplified LHS is $\left( \frac{11}{15} \right)^{9}$, which is equal to the RHS of the original given equation.

This confirms that our assumption about the missing base was correct.

The missing term in the parenthesis is $\frac{11}{15}$.

The filled-in statement is $\left( \frac{11}{15} \right)^{4}$ × $\left( \frac{11}{15} \right)^{5}$ = $\left( \frac{11}{15} \right)^{9}$.

The blank should be filled with $\frac{11}{15}$.

The given equation is $\left( \frac{-1}{4} \right)^{3} \times \left( \frac{-1}{4} \right)^{\square} = \left( \frac{-1}{4} \right)^{11}$.

Let the missing exponent in the blank be $n$. The equation is $\left( \frac{-1}{4} \right)^{3} \times \left( \frac{-1}{4} \right)^{n} = \left( \frac{-1}{4} \right)^{11}$.

We use the law of exponents for multiplication with the same base, which states that for any non-zero number $a$ and integers $m$ and $n$, $a^m \times a^n = a^{m+n}$.

In this equation, the base is $a = \frac{-1}{4}$, $m = 3$, and the unknown exponent is $n$. The equation becomes:

$\left( \frac{-1}{4} \right)^{3+n} = \left( \frac{-1}{4} \right)^{11}$

Since the bases are the same $\left( \frac{-1}{4} \right)$, the exponents must be equal.

To find $n$, subtract 3 from both sides of equation (i):

$n = 11 - 3$

$n = 8$

So, the missing exponent is 8.

The filled-in statement is $\left( \frac{-1}{4} \right)^{3}$ × $\left( \frac{-1}{4} \right)^{8}$ = $\left( \frac{-1}{4} \right)^{11}$.

Check: $\left( \frac{-1}{4} \right)^{3+8} = \left( \frac{-1}{4} \right)^{11}$. This is correct.

The blank should be filled with 8.

The given equation is $\left[ \left( \frac{7}{11} \right)^{3} \right]^4 = \left( \frac{7}{11} \right)^{\square}$.

We use the law of exponents which states that for any base $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$.

In this equation, the base is $a = \frac{7}{11}$, the inner exponent is $m = 3$, and the outer exponent is $n = 4$.

Applying the rule to the left-hand side (LHS):

$\left[ \left( \frac{7}{11} \right)^{3} \right]^4 = \left( \frac{7}{11} \right)^{3 \times 4}$

Now, perform the multiplication of the exponents:

$3 \times 4 = 12$

So, the LHS simplifies to $\left( \frac{7}{11} \right)^{12}$.

Equating the simplified LHS to the right-hand side (RHS):

$\left( \frac{7}{11} \right)^{12} = \left( \frac{7}{11} \right)^{\square}$

Since the bases are the same $\left( \frac{7}{11} \right)$, the exponents must be equal.

Therefore, the missing exponent is 12.

The filled-in statement is $\left[ \left( \frac{7}{11} \right)^{3} \right]^4$ = $\left( \frac{7}{11} \right)^{12}$.

The blank should be filled with 12.

The given equation is $\left( \frac{6}{13} \right)^{10} \div \left[ \left( \frac{6}{13} \right)^{5} \right]^2 = \left( \frac{6}{13} \right)^{\square}$.

Let the missing exponent in the blank be $p$. The equation is $\left( \frac{6}{13} \right)^{10} \div \left[ \left( \frac{6}{13} \right)^{5} \right]^2 = \left( \frac{6}{13} \right)^{p}$.

We first simplify the term inside the square brackets on the left-hand side (LHS) using the law of exponents $(a^m)^n = a^{mn}$.

$\left[ \left( \frac{6}{13} \right)^{5} \right]^2 = \left( \frac{6}{13} \right)^{5 \times 2} = \left( \frac{6}{13} \right)^{10}$

Now substitute this back into the LHS of the equation:

$LHS = \left( \frac{6}{13} \right)^{10} \div \left( \frac{6}{13} \right)^{10}$

We use the law of exponents for division with the same base, $a^m \div a^n = a^{m-n}$.

In this case, the base is $a = \frac{6}{13}$, $m = 10$, and $n = 10$.

$LHS = \left( \frac{6}{13} \right)^{10 - 10} = \left( \frac{6}{13} \right)^{0}$

Equating the simplified LHS to the RHS:

$\left( \frac{6}{13} \right)^{0} = \left( \frac{6}{13} \right)^{p}$

Since the bases are the same $\left( \frac{6}{13} \right)$, the exponents must be equal.

$0 = p$

So, the missing exponent is 0.

The filled-in statement is $\left( \frac{6}{13} \right)^{10}$ ÷ $\left[ \left( \frac{6}{13} \right)^{5} \right]^2$ = $\left( \frac{6}{13} \right)^{0}$.

The blank should be filled with 0.

The given equation is $\left[ \left( \frac{-1}{4} \right)^{16} \right]^2 = \left( \frac{-1}{4} \right)^{\square}$.

We use the law of exponents which states that for any base $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$.

In this equation, the base is $a = \frac{-1}{4}$, the inner exponent is $m = 16$, and the outer exponent is $n = 2$.

Applying the rule to the left-hand side (LHS):

$\left[ \left( \frac{-1}{4} \right)^{16} \right]^2 = \left( \frac{-1}{4} \right)^{16 \times 2}$

Now, perform the multiplication of the exponents:

$16 \times 2 = 32$

So, the LHS simplifies to $\left( \frac{-1}{4} \right)^{32}$.

Equating the simplified LHS to the right-hand side (RHS):

$\left( \frac{-1}{4} \right)^{32} = \left( \frac{-1}{4} \right)^{\square}$

Since the bases are the same $\left( \frac{-1}{4} \right)$, the exponents must be equal.

Therefore, the missing exponent is 32.

The filled-in statement is $\left[ \left( \frac{-1}{4} \right)^{16} \right]^2$ = $\left( \frac{-1}{4} \right)^{32}$.

The blank should be filled with 32.

The given equation is $\left( \frac{13}{14} \right)^{5} \div (\square)^{2} = \left( \frac{13}{14} \right)^{3}$.

Let the expression in the blank be $y$. The equation is $\left( \frac{13}{14} \right)^{5} \div (y)^{2} = \left( \frac{13}{14} \right)^{3}$.

We can rewrite the division as a fraction:

$\frac{\left( \frac{13}{14} \right)^{5}}{(y)^2} = \left( \frac{13}{14} \right)^{3}$

To solve for $(y)^2$, we can rearrange the equation. Multiply both sides by $(y)^2$ and divide both sides by $\left( \frac{13}{14} \right)^{3}$.

$(y)^2 = \frac{\left( \frac{13}{14} \right)^{5}}{\left( \frac{13}{14} \right)^{3}}$

Now, we use the law of exponents for division with the same base, which states that for any non-zero number $a$ and integers $m$ and $n$, $a^m \div a^n = a^{m-n}$.

In this case, the base is $a = \frac{13}{14}$, $m = 5$, and $n = 3$.

$(y)^2 = \left( \frac{13}{14} \right)^{5-3}$

$(y)^2 = \left( \frac{13}{14} \right)^{2}$

Taking the square root of both sides, we get $y = \pm \frac{13}{14}$. However, in the context of exponent rules and typical fill-in-the-blank questions like this, the blank is expected to be the base itself, assuming the base is $\frac{13}{14}$.

Let's assume the expression in the blank is the base $\frac{13}{14}$. Then the equation would be $\left( \frac{13}{14} \right)^{5} \div \left( \frac{13}{14} \right)^{2} = \left( \frac{13}{14} \right)^{3}$.

Using the division rule on the LHS: $\left( \frac{13}{14} \right)^{5-2} = \left( \frac{13}{14} \right)^{3}$.

$\left( \frac{13}{14} \right)^{3} = \left( \frac{13}{14} \right)^{3}$. This is true.

Therefore, the missing expression in the blank is the base $\frac{13}{14}$.

The filled-in statement is $\left( \frac{13}{14} \right)^{5}$ ÷ $\left( \frac{13}{14} \right)^{2}$ = $\left( \frac{13}{14} \right)^{3}$.

The blank should be filled with $\frac{13}{14}$.

The given equation is $a^6 \times a^5 \times a^0 = a^{\square}$.

We use the law of exponents for multiplication with the same base, which states that for any non-zero number $a$ and integers $m$, $n$, and $p$, $a^m \times a^n \times a^p = a^{m+n+p}$.

In this equation, the base is $a$, and the exponents are $6$, $5$, and $0$.

Applying the rule to the left-hand side (LHS):

$a^6 \times a^5 \times a^0 = a^{6+5+0}$

Now, perform the addition of the exponents:

$6 + 5 + 0 = 11$

So, the LHS simplifies to $a^{11}$.

Equating the simplified LHS to the right-hand side (RHS):

$a^{11} = a^{\square}$

Since the bases are the same ($a$), the exponents must be equal.

Therefore, the missing exponent is 11.

The filled-in statement is $a^6 \times a^5 \times a^0 = a^{11}$.

The blank should be filled with 11.

The given statement is 1 lakh = $10^{\square}$.

In the Indian numbering system, 1 lakh is equal to 100,000 (one hundred thousand).

We need to express 100,000 as a power of 10.

100,000 can be written as $10 \times 10 \times 10 \times 10 \times 10$.

This is the product of 10 multiplied by itself 5 times.

So, $100,000 = 10^5$.

Substitute this value back into the given statement:

$10^5 = 10^{\square}$

Since the bases are the same (10), the exponents must be equal.

Therefore, the missing exponent is 5.

The filled-in statement is 1 lakh = $10^5$.

The blank should be filled with 5.

The given statement is 1 million = $10^{\square}$.

In the international numbering system, 1 million is equal to 1,000,000 (one million).

We need to express 1,000,000 as a power of 10.

1,000,000 can be written as $10 \times 10 \times 10 \times 10 \times 10 \times 10$.

This is the product of 10 multiplied by itself 6 times.

So, $1,000,000 = 10^6$.

Substitute this value back into the given statement:

$10^6 = 10^{\square}$

Since the bases are the same (10), the exponents must be equal.

Therefore, the missing exponent is 6.

The filled-in statement is 1 million = $10^6$.

The blank should be filled with 6.

Given:

The equation is $729 = 3^{-}$.

To Find:

The missing exponent in the given equation.

Solution:

To find the missing exponent, we need to express the number 729 as a power of 3. We can achieve this by performing the prime factorization of 729 using the prime number 3.

The prime factorization of 729 is shown below:

$\begin{array}{c|cc} 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$

From the prime factorization diagram, we observe that 729 is obtained by multiplying 3 by itself 6 times.

$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$

In exponential form, this product is written as:

$729 = 3^6$

Now, we compare this result with the given equation:

$3^6 = 3^{-}$

By comparing the exponents on both sides of the equality, we can determine the value of the missing exponent. The exponent on the left side is 6.

Therefore, the missing exponent on the right side must also be 6.

The blank should be filled with 6.

Given:

The equation is $432 = 2^4 \times 3^{-}$.

To Find:

The missing exponent of 3 in the given equation.

Solution:

We are given the equation $432 = 2^4 \times 3^{x}$, where $x$ is the missing exponent we need to find.

First, let's calculate the value of $2^4$:

$2^4 = 2 \times 2 \times 2 \times 2 = 16$.

The equation now becomes:

$432 = 16 \times 3^x$

To find $3^x$, we need to divide 432 by 16:

$3^x = \frac{432}{16}$

Now, perform the division:

$\frac{432}{16} = 27$

So, we have:

$3^x = 27$

To find the value of $x$, we need to express 27 as a power of 3. We can do this by finding the prime factorization of 27 using the base 3.

$27 = 3 \times 3 \times 3$

In exponential form, this is $3^3$.

So, the equation $3^x = 27$ becomes:

$3^x = 3^3$

Since the bases are the same (both are 3), the exponents must be equal.

Therefore, $x = 3$.

The missing exponent is 3.

The blank should be filled with 3.

Alternate Solution (using complete prime factorization):

Find the prime factorization of 432:

$\begin{array}{c|cc} 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$

So, $432 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$.

Writing this in exponential form:

$432 = 2^4 \times 3^3$

Comparing this with the given equation $432 = 2^4 \times 3^{-}$, we can see that the missing exponent of 3 is 3.

The blank should be filled with 3.

Given:

The equation is $53700000 = \text{___} \times 10^7$.

To Find:

The missing number in the given equation.

Solution:

We need to express the number 53700000 in the form of $a \times 10^7$, where $a$ is the number that fills the blank.

The number is 53,700,000.

The power of 10 required is $10^7$. This means we need to shift the decimal point 7 places to the left from its original position (which is implicitly after the last digit in a whole number).

Let's start with 53700000.

Moving the decimal point 7 places to the left:

1st place: 5370000.0

2nd place: 537000.0

3rd place: 53700.0

4th place: 5370.0

5th place: 537.0

6th place: 53.7

7th place: 5.37

So, $53700000 = 5.37 \times 10^7$.

Comparing this with the given equation $53700000 = \text{___} \times 10^7$, we see that the missing number is 5.37.

Alternatively, let the missing number be $x$. The equation is:

$53700000 = x \times 10^7$

To find $x$, we divide both sides by $10^7$:

$x = \frac{53700000}{10^7}$

$10^7 = 10,000,000$.

$x = \frac{53700000}{10000000}$

Dividing 53700000 by 10000000 gives:

$x = 5.37$

Thus, the missing number is 5.37.

The blank should be filled with 5.37.

Given:

The equation is $88880000000 = \text{___} \times 10^{10}$.

To Find:

The missing number in the given equation.

Solution:

We need to express the number 88880000000 in the form of $x \times 10^{10}$, where $x$ is the number that fills the blank.

Let the missing number be $x$. The equation is:

$88880000000 = x \times 10^{10}$

To find the value of $x$, we need to divide 88880000000 by $10^{10}$.

$x = \frac{88880000000}{10^{10}}$

The value of $10^{10}$ is 1 followed by 10 zeros, which is 10,000,000,000.

So, we need to calculate:

$x = \frac{88880000000}{10000000000}$

Dividing a number by $10^{10}$ is equivalent to moving the decimal point 10 places to the left from its current position. In the number 88880000000, the decimal point is implicitly after the last zero.

Moving the decimal point 10 places to the left:

Starting from 88880000000.

After 10 shifts to the left, the decimal point will be between the first 8 and the second 8.

88880000000 becomes 8.888.

So, $x = 8.888$.

Therefore, the missing number is 8.888.

The equation is $88880000000 = 8.888 \times 10^{10}$.

The blank should be filled with 8.888.

Given:

The equation is $27500000 = 2.75 \times 10^{-}$.

To Find:

The missing exponent in the given equation.

Solution:

We are asked to fill the blank in the equation $27500000 = 2.75 \times 10^{x}$, where $x$ is the missing exponent.

The number 27,500,000 is a large number. We need to express it as a product of 2.75 and a power of 10.

Consider the number 2.75. To get 27,500,000 from 2.75, we need to multiply 2.75 by a certain power of 10. This is equivalent to shifting the decimal point in 2.75 to the right.

The number 27,500,000 can be written with the decimal point at the end as 27500000.0.

In 2.75, the decimal point is after the digit 2. To move the decimal point from its position in 2.75 to its position in 27500000.0, we count the number of places the decimal point needs to move to the right.

From 2.75, move the decimal point right:

• 1 place: 27.5
• 2 places: 275.
• 3 places: 2750.
• 4 places: 27500.
• 5 places: 275000.
• 6 places: 2750000.
• 7 places: 27500000.

The decimal point is moved 7 places to the right.

Moving the decimal point $n$ places to the right is equivalent to multiplying by $10^n$.

So, $27500000 = 2.75 \times 10^7$.

Comparing this with the given equation $27500000 = 2.75 \times 10^{-}$, we find that the missing exponent is 7.

Alternatively, let the missing exponent be $x$.

$27500000 = 2.75 \times 10^x$

Divide both sides by 2.75 to find $10^x$:

$10^x = \frac{27500000}{2.75}$

We can perform the division:

$\frac{27500000}{2.75} = \frac{27500000}{\frac{275}{100}} = 27500000 \times \frac{100}{275}$

$\frac{27500000}{275} = \frac{275 \times 100000}{275} = 100000$

So, $10^x = 100000 \times 100 = 10000000$.

We need to write 10,000,000 as a power of 10.

$10000000 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^7$

Thus, $10^x = 10^7$.

Since the bases are equal, the exponents must be equal.

$x = 7$.

The missing exponent is 7.

The blank should be filled with 7.

Given:

The equation is $340900000 = 3.409 \times 10^{-}$.

To Find:

The missing exponent in the given equation.

Solution:

We need to determine the exponent $x$ such that $340900000 = 3.409 \times 10^x$.

The number 340,900,000 is a large number. We are expressing it in a form similar to scientific notation, where a number between 1 and 10 is multiplied by a power of 10. In this case, the number is 3.409.

To convert 340,900,000 into the form $3.409 \times 10^x$, we need to see how many places the decimal point needs to be moved from its implicit position in 340900000 (which is after the last zero) to arrive at the position in 3.409 (after the digit 3).

Let's count the number of places we move the decimal point to the left from the end of 340900000:

$340900000.$

Moving left:

1 place: 34090000.0

2 places: 3409000.00

3 places: 340900.000

4 places: 34090.0000

5 places: 3409.00000

6 places: 340.900000

7 places: 34.0900000

8 places: 3.40900000

The decimal point is moved 8 places to the left. Moving the decimal point $n$ places to the left is equivalent to multiplying by $10^n$.

So, $340900000 = 3.409 \times 10^8$.

Comparing this with the given equation $340900000 = 3.409 \times 10^{-}$, we see that the missing exponent is 8.

Alternatively, let the missing exponent be $x$.

$340900000 = 3.409 \times 10^x$

Divide both sides by 3.409:

$10^x = \frac{340900000}{3.409}$

To simplify the division, we can write 3.409 as $\frac{3409}{1000}$.

$10^x = \frac{340900000}{\frac{3409}{1000}} = 340900000 \times \frac{1000}{3409}$

Since $340900000 = 3409 \times 100000$, we have:

$10^x = \frac{3409 \times 100000 \times 1000}{3409}$

Cancel out 3409 from the numerator and denominator:

$10^x = 100000 \times 1000$

$100000 \times 1000 = 10^5 \times 10^3 = 10^{5+3} = 10^8$.

So, $10^x = 10^8$.

Since the bases are equal, the exponents must be equal.

$x = 8$.

The missing exponent is 8.

The blank should be filled with 8.

Given:

Several pairs of numbers, some in exponential form, to be compared.

To Find:

The correct comparison sign ($<$, $>$, or $=$) that makes each statement true.

Solution:

To compare the numbers, we first need to evaluate the expressions on both sides of the blank for each pair.

(a) $3^2$ ______ $15$

Evaluate the exponential term:

$3^2 = 3 \times 3 = 9$

Now, compare 9 and 15:

$9 < 15$

So, the correct sign is $<$.

$3^2 \ \mathbf{<} \ 15$

(b) $2^3$ ______ $3^2$

Evaluate the terms:

$2^3 = 2 \times 2 \times 2 = 8$

$3^2 = 3 \times 3 = 9$

Now, compare 8 and 9:

$8 < 9$

So, the correct sign is $<$.

$2^3 \ \mathbf{<} \ 3^2$

(c) $7^4$ ______ $5^4$

Evaluate the terms:

$7^4 = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2401$

$5^4 = 5 \times 5 \times 5 \times 5 = 25 \times 25 = 625$

Now, compare 2401 and 625:

$2401 > 625$

So, the correct sign is $>$.

$7^4 \ \mathbf{>} \ 5^4$

(d) $10,000$ ______ $10^5$

Evaluate the exponential term:

$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000$

Now, compare 10,000 and 100,000:

$10,000 < 100,000$

So, the correct sign is $<$.

$10,000 \ \mathbf{<} \ 10^5$

(e) $6^3$ _____ $4^4$

Evaluate the terms:

$6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$

$4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256$

Now, compare 216 and 256:

$216 < 256$

So, the correct sign is $<$.

$6^3 \ \mathbf{<} \ 4^4$

Final Answers:

(a) $3^2$ $\mathbf{<}$ $15$

(b) $2^3$ $\mathbf{<}$ $3^2$

(c) $7^4$ $\mathbf{>}$ $5^4$

(d) $10,000$ $\mathbf{<}$ $10^5$

(e) $6^3$ $\mathbf{<}$ $4^4$

Given:

The statement is "One million $= 10^7$".

To Determine:

Whether the given statement is True or False.

Solution:

We need to compare the value of one million with the value of $10^7$.

One million is written numerically as 1,000,000.

In terms of powers of 10, 1,000,000 has 6 zeros after the digit 1. So, $1,000,000 = 10^6$.

Now let's evaluate the right side of the given statement, which is $10^7$.

$10^7$ represents 10 raised to the power of 7. This means 1 followed by 7 zeros.

$10^7 = 10,000,000$.

The statement claims that one million is equal to $10^7$.

This is equivalent to claiming that $1,000,000 = 10,000,000$.

Comparing the two numbers, we can clearly see that 1,000,000 is not equal to 10,000,000.

Therefore, the given statement "One million $= 10^7$" is not true.

The statement is False.

Given:

The statement is "One hour $= 60^2$ seconds".

To Determine:

Whether the given statement is True or False.

Solution:

We need to check if the duration of one hour is equal to the value of $60^2$ seconds.

First, let's convert one hour into seconds.

We know that:

1 hour = 60 minutes

And:

1 minute = 60 seconds

To find the number of seconds in one hour, we multiply the number of minutes in an hour by the number of seconds in a minute.

Number of seconds in one hour $= (\text{Number of minutes in 1 hour}) \times (\text{Number of seconds in 1 minute})$

Number of seconds in one hour $= 60 \times 60$ seconds

The expression $60 \times 60$ can be written in exponential form as $60^2$.

So, 1 hour $= 60^2$ seconds.

Now, let's evaluate $60^2$:

$60^2 = 60 \times 60 = 3600$

So, 1 hour $= 3600$ seconds.

The given statement is "One hour $= 60^2$ seconds". Since $60^2$ seconds is indeed equal to one hour, the statement is correct.

Therefore, the statement is True.

Given:

The statement is "$1^0 \times 0^1 = 1$".

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the left side of the equation, $1^0 \times 0^1$, and see if it equals the right side, 1.

First, evaluate the term $1^0$.

Any non-zero number raised to the power of 0 is equal to 1. In this case, the base is 1, which is non-zero.

So, $1^0 = 1$.

Next, evaluate the term $0^1$.

Any number raised to the power of 1 is equal to the base itself. In this case, the base is 0.

So, $0^1 = 0$.

Now, substitute these values back into the expression on the left side of the given statement:

$1^0 \times 0^1 = 1 \times 0$

Perform the multiplication:

$1 \times 0 = 0$

The left side of the statement evaluates to 0.

Now, we compare this result with the right side of the statement, which is 1.

$0 = 1$

This equality is false.

Therefore, the given statement "$1^0 \times 0^1 = 1$" is incorrect.

The statement is False.

Given:

The statement is "$(-3)^4 = -12$".

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the left side of the equation, $(-3)^4$, and compare it with the right side, $-12$.

The expression $(-3)^4$ means $(-3)$ multiplied by itself 4 times.

$(-3)^4 = (-3) \times (-3) \times (-3) \times (-3)$

Let's multiply the terms step by step:

$(-3) \times (-3) = 9$

Now, multiply the result by the next $(-3)$:

$9 \times (-3) = -27$

Finally, multiply this result by the last $(-3)$:

$-27 \times (-3) = 81$

So, $(-3)^4 = 81$.

Now, we compare the calculated value of $(-3)^4$ with the value on the right side of the given statement:

$81 = -12$

This equality is false. The number 81 is positive and much larger than the negative number -12.

Therefore, the given statement "$(-3)^4 = -12$" is incorrect.

The statement is False.

Given:

The statement is "$3^4 > 4^3$".

To Determine:

Whether the given statement is True or False.

Solution:

To determine if the statement is true, we need to evaluate the expressions on both sides of the inequality and compare their values.

First, evaluate the left side, $3^4$:

$3^4 = 3 \times 3 \times 3 \times 3$

$3 \times 3 = 9$

$9 \times 3 = 27$

$27 \times 3 = 81$

So, $3^4 = 81$.

Next, evaluate the right side, $4^3$:

$4^3 = 4 \times 4 \times 4$

$4 \times 4 = 16$

$16 \times 4 = 64$

So, $4^3 = 64$.

Now, we substitute the calculated values back into the inequality:

$81 > 64$

This inequality is true because 81 is indeed greater than 64.

Therefore, the given statement "$3^4 > 4^3$" is correct.

The statement is True.

Given:

The statement is $\left( \frac{-3}{5} \right)^{100} = \frac{-3^{100}}{-5^{100}}$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the equation and check if they are equal.

Consider the left side of the equation: $\left( \frac{-3}{5} \right)^{100}$.

Using the property of exponents $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$, we can write:

$\left( \frac{-3}{5} \right)^{100} = \frac{(-3)^{100}}{5^{100}}$

For the term $(-3)^{100}$, since the base is negative and the exponent (100) is an even number, the result will be positive.

$(-3)^{100} = (-1 \times 3)^{100} = (-1)^{100} \times 3^{100}$.

Since $(-1)^{100} = 1$ (because 100 is even), we have:

$(-3)^{100} = 1 \times 3^{100} = 3^{100}$.

The term $5^{100}$ is $5 \times 5 \times ...$ (100 times), which is a positive number.

So, the left side simplifies to:

$\left( \frac{-3}{5} \right)^{100} = \frac{3^{100}}{5^{100}}$.

Now consider the right side of the equation: $\frac{-3^{100}}{-5^{100}}$.

The term $3^{100}$ is positive. So, $-3^{100}$ is a negative number.

The term $5^{100}$ is positive. So, $-5^{100}$ is a negative number.

When a negative number is divided by a negative number, the result is a positive number.

$\frac{-3^{100}}{-5^{100}} = \frac{-(3^{100})}{-(5^{100})}$

The negative signs cancel out:

$\frac{-(3^{100})}{-(5^{100})} = \frac{3^{100}}{5^{100}}$.

Comparing the simplified left side and the simplified right side:

Left side: $\frac{3^{100}}{5^{100}}$

Right side: $\frac{3^{100}}{5^{100}}$

Since both sides are equal, the given statement is true.

Therefore, the statement is True.

Given:

The statement is $(10 + 10)^{10} = 10^{10} + 10^{10}$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the equation and check if they are equal.

Consider the left side of the equation: $(10 + 10)^{10}$.

First, perform the addition inside the parentheses:

$10 + 10 = 20$

So, the left side is $(20)^{10}$.

$(20)^{10} = (2 \times 10)^{10}$

Using the property of exponents $(ab)^m = a^m b^m$, we can write:

$(2 \times 10)^{10} = 2^{10} \times 10^{10}$

Now, consider the right side of the equation: $10^{10} + 10^{10}$.

This is a sum of two identical terms. We can factor out the common term $10^{10}$:

$10^{10} + 10^{10} = 1 \times 10^{10} + 1 \times 10^{10} = (1 + 1) \times 10^{10}$

$(1 + 1) \times 10^{10} = 2 \times 10^{10}$

Now, we compare the simplified left side and the simplified right side:

Left side: $2^{10} \times 10^{10}$

Right side: $2 \times 10^{10}$

To see if they are equal, we can compare $2^{10}$ and $2$.

$2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024$.

So, the left side is $1024 \times 10^{10}$.

The right side is $2 \times 10^{10}$.

Since $1024 \neq 2$, the left side $1024 \times 10^{10}$ is not equal to the right side $2 \times 10^{10}$.

In general, $(a+b)^m \neq a^m + b^m$ unless $m=1$ or $a$ or $b$ (or both) are zero.

Therefore, the given statement is incorrect.

The statement is False.

Given:

The statement is "$x^0 \times x^0 = x^0 \div x^0$" is true for all non-zero values of $x$.

To Determine:

Whether the given statement is True or False.

Solution:

The statement claims that the equation $x^0 \times x^0 = x^0 \div x^0$ holds true for any value of $x$ that is not equal to zero ($x \neq 0$).

Let's analyze the term $x^0$ for $x \neq 0$.

By definition, any non-zero number raised to the power of 0 is equal to 1.

So, for any $x \neq 0$, we have $x^0 = 1$.

Now, let's substitute $x^0 = 1$ into the given equation:

The left side (LHS) of the equation is $x^0 \times x^0$.

LHS $= 1 \times 1$

LHS $= 1$

The right side (RHS) of the equation is $x^0 \div x^0$.

RHS $= 1 \div 1$

RHS $= 1$

Comparing the left side and the right side, we have:

$1 = 1$

The equality holds true.

Since $x^0 = 1$ is valid for all non-zero values of $x$, the equation $x^0 \times x^0 = x^0 \div x^0$ simplifies to $1 \times 1 = 1 \div 1$, which is $1 = 1$. This is true for all non-zero $x$.

Note that the division $x^0 \div x^0$ is valid because for $x \neq 0$, $x^0 = 1$, and division by 1 is allowed.

Therefore, the given statement is correct.

The statement is True.

Given:

The statement is "In the standard form, a large number can be expressed as a decimal number between 0 and 1, multiplied by a power of 10."

To Determine:

Whether the given statement is True or False.

Solution:

The standard form of a number (also known as scientific notation) is a way to write very large or very small numbers concisely. A number is written in standard form as:

$a \times 10^n$

where $a$ is a decimal number (called the mantissa or significand) and $n$ is an integer (positive or negative).

The rule for the standard form is that the absolute value of $a$ must be greater than or equal to 1 and less than 10.

$1 \leq |a| < 10$

For a large positive number, $a$ will be a positive decimal number such that $1 \leq a < 10$, and the exponent $n$ will be a positive integer.

The given statement says that the decimal number is "between 0 and 1". This means $0 < a < 1$.

This condition ($0 < a < 1$) is different from the condition required for standard form ($1 \leq a < 10$).

For example, consider the large number 5,000.

In standard form, this is written as $5 \times 10^3$. Here, the decimal number is 5, which satisfies $1 \leq 5 < 10$.

If we were to express 5,000 with a decimal number between 0 and 1, we would write it as $0.5 \times 10^4$. Here, the decimal number 0.5 is between 0 and 1, but this is not the standard form according to the widely accepted definition.

The standard convention requires the coefficient to be $1 \leq |a| < 10$.

Therefore, the statement contradicts the definition of the standard form for a large number.

The statement is False.

Given:

The statement is "$4^2$ is greater than $2^4$". This can be written as the inequality $4^2 > 2^4$.

To Determine:

Whether the given statement is True or False.

Solution:

To determine if the statement is true, we need to evaluate the expressions on both sides of the inequality and compare their values.

First, evaluate the left side, $4^2$:

$4^2 = 4 \times 4 = 16$.

So, the value of $4^2$ is 16.

Next, evaluate the right side, $2^4$:

$2^4 = 2 \times 2 \times 2 \times 2$.

$2 \times 2 = 4$

$4 \times 2 = 8$

$8 \times 2 = 16$

So, the value of $2^4$ is 16.

Now, we substitute the calculated values back into the original statement's inequality:

$16 > 16$

This inequality states that 16 is greater than 16, which is false. 16 is equal to 16.

Therefore, the given statement "$4^2$ is greater than $2^4$" is incorrect.

The statement is False.

Given:

The statement is "$x^m + x^m = x^{2m}$", where $x$ is a non-zero rational number and $m$ is a positive integer.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the equation and see if they are equal for all non-zero rational numbers $x$ and all positive integers $m$.

Consider the left side (LHS) of the equation: $x^m + x^m$.

This is the sum of two identical terms, $x^m$. We can combine these terms by addition:

$x^m + x^m = 1 \cdot x^m + 1 \cdot x^m = (1 + 1) \cdot x^m = 2 \cdot x^m$

So, LHS $= 2x^m$.

Now consider the right side (RHS) of the equation: $x^{2m}$.

Using the property of exponents $(a^n)^p = a^{np}$, we can also write $x^{2m}$ as $(x^m)^2$.

RHS $= x^{2m} = (x^m)^2$.

The statement claims that $2x^m = x^{2m}$ for all non-zero rational $x$ and positive integer $m$.

Let's test this with an example.

Let $x = 3$ and $m = 2$.

LHS $= 3^2 + 3^2 = 9 + 9 = 18$.

RHS $= 3^{2 \times 2} = 3^4 = 3 \times 3 \times 3 \times 3 = 81$.

In this case, $18 \neq 81$. So the statement is false for $x=3, m=2$.

For the equality $2x^m = x^{2m}$ to hold, we would need $2x^m = (x^m)^2$.

Let $y = x^m$. The equation becomes $2y = y^2$.

Rearranging gives $y^2 - 2y = 0$, which factors as $y(y - 2) = 0$.

This equation is true if $y = 0$ or $y - 2 = 0$, i.e., $y = 0$ or $y = 2$.

Substituting back $y = x^m$, the original equation is true if $x^m = 0$ or $x^m = 2$.

Since $m$ is a positive integer, $x^m = 0$ implies $x=0$. However, the statement is given for non-zero values of $x$.

The equation is true if $x^m = 2$. This is only true for specific combinations of $x$ and $m$ (e.g., if $m=1$, $x=2$; if $m=3$, $x = \sqrt[3]{2}$), but not for all non-zero rational $x$ and all positive integers $m$.

Since we found a counterexample ($x=3, m=2$) where the equality does not hold, the statement is false.

Therefore, the statement is False.

Given:

The statement is "$x^m \times y^m = (x \times y)^{2m}$", where $x$ and $y$ are non-zero rational numbers and $m$ is a positive integer.

To Determine:

Whether the given statement is True or False.

Solution:

We need to check if the equality $x^m \times y^m = (x \times y)^{2m}$ holds true for all non-zero rational numbers $x, y$ and all positive integers $m$.

Consider the left side (LHS) of the equation: $x^m \times y^m$.

This expression involves the product of two powers with different bases ($x$ and $y$) but the same exponent ($m$). Using the property of exponents which states that $a^m \times b^m = (a \times b)^m$, we can simplify the LHS:

LHS $= x^m \times y^m = (x \times y)^m$.

Now, consider the right side (RHS) of the equation: $(x \times y)^{2m}$.

The given statement claims that the LHS is equal to the RHS:

$(x \times y)^m = (x \times y)^{2m}$

Let $z = x \times y$. Since $x$ and $y$ are non-zero, their product $z$ is also non-zero ($z \neq 0$). The equation becomes:

$z^m = z^{2m}$

We can rewrite $z^{2m}$ as $(z^m)^2$ using the property $(a^n)^p = a^{np}$.

$z^m = (z^m)^2$

Let $w = z^m$. Since $z \neq 0$ and $m$ is a positive integer, $w = z^m$ is also non-zero ($w \neq 0$). The equation is:

$w = w^2$

To solve for $w$, rearrange the equation:

$w^2 - w = 0$

Factor out $w$:

$w(w - 1) = 0$

This equation is true if $w = 0$ or $w - 1 = 0$.

So, $w = 0$ or $w = 1$.

Substituting back $w = z^m$, the original equality $z^m = z^{2m}$ is true if $z^m = 0$ or $z^m = 1$.

Since $z = x \times y$ is non-zero and $m$ is a positive integer, $z^m$ cannot be 0. Thus, the only possibility for the equality to hold is $z^m = 1$.

$z^m = 1 \implies (x \times y)^m = 1$.

This equality $(x \times y)^m = 1$ is only true under specific conditions, such as when $x \times y = 1$ (for any positive integer $m$) or when $x \times y = -1$ and $m$ is an even positive integer. It is not true for all non-zero rational numbers $x, y$ and all positive integers $m$.

For example, let $x = 2$, $y = 1$, and $m = 2$.

LHS $= x^m \times y^m = 2^2 \times 1^2 = 4 \times 1 = 4$.

RHS $= (x \times y)^{2m} = (2 \times 1)^{2 \times 2} = 2^4 = 16$.

Since $4 \neq 16$, the statement is false for these values.

The correct property is $x^m \times y^m = (x \times y)^m$.

Therefore, the given statement "$x^m \times y^m = (x \times y)^{2m}$" is incorrect.

The statement is False.

Given:

The statement is "$x^m \div y^m = (x \div y)^m$", where $x$ and $y$ are non-zero rational numbers and $m$ is a positive integer.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the truthfulness of the equation $x^m \div y^m = (x \div y)^m$ under the given conditions ($x \neq 0$, $y \neq 0$, $m$ is a positive integer).

The equation involves division of powers with the same exponent $m$ but different bases $x$ and $y$.

We can rewrite the division using a fraction bar:

$\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$

This is a fundamental property of exponents, which states that the quotient of powers with the same exponent is equal to the quotient of the bases raised to that exponent. This property is valid as long as the bases $x$ and $y$ are non-zero and $y$ is not zero (since it is in the denominator), and $m$ is any integer (for non-zero bases).

The conditions given in the statement are that $x$ and $y$ are non-zero rational numbers, and $m$ is a positive integer. These conditions satisfy the requirements for the property to be true ($x \neq 0$ and $y \neq 0$).

Therefore, the equality $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$, which is equivalent to $x^m \div y^m = (x \div y)^m$, is true under the specified conditions.

For example, let $x = 4$, $y = 2$, and $m = 3$.

LHS $= x^m \div y^m = 4^3 \div 2^3 = 64 \div 8 = 8$.

RHS $= (x \div y)^m = (4 \div 2)^3 = 2^3 = 8$.

Here, LHS $=$ RHS.

Let $x = -6$, $y = 3$, and $m = 2$.

LHS $= x^m \div y^m = (-6)^2 \div 3^2 = 36 \div 9 = 4$.

RHS $= (x \div y)^m = (-6 \div 3)^2 = (-2)^2 = 4$.

Here again, LHS $=$ RHS.

The property holds true for all non-zero rational $x, y$ and positive integer $m$.

Therefore, the statement is True.

Given:

The statement is "$x^m \times x^n = x^{m + n}$", where $x$ is a non-zero rational number and $m, n$ are positive integers.

To Determine:

Whether the given statement is True or False.

Solution:

The given statement is an equation involving exponents. It describes a rule for multiplying powers with the same base.

The rule for multiplying powers with the same base states that when you multiply two exponential terms with the same base, you keep the base and add the exponents. Mathematically, this is expressed as:

$a^p \times a^q = a^{p+q}$

In the given statement, the base is $x$, the first exponent is $m$, and the second exponent is $n$. The statement claims that:

$x^m \times x^n = x^{m+n}$

This matches the standard rule for multiplying powers with the same base.

The rule $a^p \times a^q = a^{p+q}$ is valid when the base $a$ is any non-zero number and the exponents $p$ and $q$ are integers. In this statement, $x$ is given as a non-zero rational number, which is a valid base. The exponents $m$ and $n$ are given as positive integers, which are also valid exponents for this rule.

Since the given equation is the correct rule for multiplying exponents with the same base under the specified conditions, the statement is true.

Therefore, the statement is True.

Given:

The statement is "$4^9$ is greater than $16^3$". This can be written as the inequality $4^9 > 16^3$.

To Determine:

Whether the given statement is True or False.

Solution:

To determine if the inequality $4^9 > 16^3$ is true, we can either evaluate both sides or express them with the same base and compare the exponents. It is generally easier to express both sides with the same base.

The bases are 4 and 16. We notice that 16 can be expressed as a power of 4, since $16 = 4 \times 4 = 4^2$.

The left side of the inequality is $4^9$.

The right side of the inequality is $16^3$. Substitute $16 = 4^2$:

$16^3 = (4^2)^3$

Using the exponent property $(a^m)^n = a^{mn}$, we multiply the exponents:

$(4^2)^3 = 4^{2 \times 3} = 4^6$

Now, the original inequality $4^9 > 16^3$ becomes:

$4^9 > 4^6$

When comparing two powers with the same base greater than 1, the power with the larger exponent is the greater number.

In this case, the base is 4, which is greater than 1. The exponents are 9 and 6. Since $9 > 6$, it follows that $4^9 > 4^6$.

Therefore, the inequality $4^9 > 16^3$ is true.

Alternatively, we can evaluate both sides:

$4^9 = 262144$

$16^3 = 16 \times 16 \times 16 = 256 \times 16 = 4096$

Comparing the values, $262144 > 4096$, which confirms the inequality is true.

Thus, the given statement is correct.

The statement is True.

Given:

The statement is $\left( \frac{2}{5} \right)^{3} \div \left( \frac{5}{2} \right)^{3} = 1$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the left side of the equation and compare it to the right side (which is 1).

The left side of the equation is $\left( \frac{2}{5} \right)^{3} \div \left( \frac{5}{2} \right)^{3}$.

We are dividing two exponential terms that have the same exponent (3) but different bases ($\frac{2}{5}$ and $\frac{5}{2}$).

We can use the property of exponents that states $a^m \div b^m = (a \div b)^m$ for non-zero bases $a$ and $b$.

Applying this property to the left side:

LHS $= \left( \frac{2}{5} \div \frac{5}{2} \right)^{3}$

Now, we perform the division inside the parentheses. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{5}{2}$ is $\frac{2}{5}$.

$\frac{2}{5} \div \frac{5}{2} = \frac{2}{5} \times \frac{2}{5} = \left(\frac{2}{5}\right)^2$

So, the left side becomes:

LHS $= \left( \left(\frac{2}{5}\right)^2 \right)^3$

Using the property of exponents $(a^m)^n = a^{mn}$, we multiply the exponents:

LHS $= \left(\frac{2}{5}\right)^{2 \times 3} = \left(\frac{2}{5}\right)^6$

Now, we evaluate $\left(\frac{2}{5}\right)^6$:

$\left(\frac{2}{5}\right)^6 = \frac{2^6}{5^6} = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 5 \times 5 \times 5} = \frac{64}{15625}$

The left side of the given statement is $\frac{64}{15625}$.

The right side of the given statement is 1.

Comparing the two values:

$\frac{64}{15625} = 1$

This equality is false, since $\frac{64}{15625}$ is a fraction less than 1.

Therefore, the given statement is incorrect.

Alternate Solution:

Consider the left side: $\left( \frac{2}{5} \right)^{3} \div \left( \frac{5}{2} \right)^{3}$.

We can write the division as a fraction:

LHS $= \frac{\left( \frac{2}{5} \right)^{3}}{\left( \frac{5}{2} \right)^{3}}$

Using the property $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$:

LHS $= \frac{\frac{2^3}{5^3}}{\frac{5^3}{2^3}}$

To divide by a fraction, multiply by its reciprocal:

LHS $= \frac{2^3}{5^3} \times \frac{2^3}{5^3}$

Multiply the numerators and the denominators:

LHS $= \frac{2^3 \times 2^3}{5^3 \times 5^3}$

Using the property $a^m \times a^n = a^{m+n}$:

LHS $= \frac{2^{3+3}}{5^{3+3}} = \frac{2^6}{5^6}$

LHS $= \left(\frac{2}{5}\right)^6 = \frac{64}{15625}$

Comparing with the right side, $1$:

$\frac{64}{15625} \neq 1$

The statement is false.

Therefore, the statement is False.

Given:

The statement is $\left( \frac{4}{3} \right)^{5} \times \left( \frac{5}{7} \right)^{3} = \left( \frac{4}{3} \;+\; \frac{5}{7} \right)^{5}$.

To Determine:

Whether the given statement is True or False.

Solution:

The statement claims that a product of two powers with different bases and different exponents is equal to the power of the sum of the bases.

Let's recall the standard rules of exponents:

• $a^m \times a^n = a^{m+n}$ (same base, different exponents)
• $a^m \times b^m = (a \times b)^m$ (different bases, same exponent)
• $(a^m)^n = a^{mn}$ (power of a power)

The given statement, $\left( \frac{4}{3} \right)^{5} \times \left( \frac{5}{7} \right)^{3} = \left( \frac{4}{3} \;+\; \frac{5}{7} \right)^{5}$, does not match any of these standard rules or any other valid exponent rule. Exponent rules apply to multiplication, division, and powers of powers, not directly to sums raised to powers or products of powers with different bases and exponents combined in this manner.

In general, $(a+b)^m$ is not equal to $a^m \times b^m$ or $a^m \times b^n$ for different $m, n$.

Consider a simple example with positive integers:

Let $a=2$, $b=3$, $m=2$, $n=1$.

The statement structure would suggest something like $a^m \times b^n = (a+b)^m$.

Let's check if $2^2 \times 3^1 = (2+3)^2$.

Left side: $2^2 \times 3^1 = 4 \times 3 = 12$.

Right side: $(2+3)^2 = 5^2 = 25$.

Since $12 \neq 25$, this shows the general principle is false.

Similarly, for the given statement, $\left( \frac{4}{3} \right)^{5} \times \left( \frac{5}{7} \right)^{3}$ will result in a specific value, and $\left( \frac{4}{3} \;+\; \frac{5}{7} \right)^{5} = \left( \frac{43}{21} \right)^{5}$ will result in a very different value.

The given equality is not based on any valid mathematical property.

Therefore, the statement is False.

Given:

The statement is $\left( \frac{5}{8} \right)^{9} \div \left( \frac{5}{8} \right)^{4} = \left( \frac{5}{8} \right)^{4}$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to simplify the left side of the equation using the rules of exponents and compare it to the right side.

The left side of the equation is $\left( \frac{5}{8} \right)^{9} \div \left( \frac{5}{8} \right)^{4}$.

This is a division of powers with the same base ($\frac{5}{8}$). The rule for dividing powers with the same base is $a^m \div a^n = a^{m-n}$.

Applying this rule to the left side:

LHS $= \left( \frac{5}{8} \right)^{9 - 4}$

Calculate the difference in exponents:

$9 - 4 = 5$

So, the left side simplifies to:

LHS $= \left( \frac{5}{8} \right)^{5}$

Now, compare the simplified left side with the right side of the given statement:

$\left( \frac{5}{8} \right)^{5} = \left( \frac{5}{8} \right)^{4}$

This equality is true only if the base is 0, 1, or -1 under certain conditions, or if the exponents are equal.

The base here is $\frac{5}{8}$, which is not 0, 1, or -1.

The exponents are 5 and 4. Since $5 \neq 4$, the equality $\left( \frac{5}{8} \right)^{5} = \left( \frac{5}{8} \right)^{4}$ is false.

Therefore, the original statement is incorrect.

The statement is False.

Given:

The statement is $\left( \frac{7}{3} \right)^{2} \times \left( \frac{7}{3} \right)^{5} = \left( \frac{7}{3} \right)^{10}$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the left side of the equation using the rules of exponents and compare the result to the right side.

The left side of the equation is $\left( \frac{7}{3} \right)^{2} \times \left( \frac{7}{3} \right)^{5}$.

This is a product of two exponential terms that have the same base, which is $\frac{7}{3}$.

The rule for multiplying powers with the same base states that we add the exponents: $a^m \times a^n = a^{m+n}$.

Applying this rule to the left side:

LHS $= \left( \frac{7}{3} \right)^{2+5}$

Calculate the sum of the exponents:

$2 + 5 = 7$

So, the left side simplifies to:

LHS $= \left( \frac{7}{3} \right)^{7}$

Now, we compare the simplified left side with the right side of the given statement:

$\left( \frac{7}{3} \right)^{7} = \left( \frac{7}{3} \right)^{10}$

For this equality to be true, since the bases are the same ($\frac{7}{3}$ which is not 0 or 1 or -1), the exponents must be equal.

Comparing the exponents 7 and 10, we see that $7 \neq 10$.

Therefore, the equality $\left( \frac{7}{3} \right)^{7} = \left( \frac{7}{3} \right)^{10}$ is false.

The given statement is incorrect.

The statement is False.

Given:

The statement is $5^0 \times 25^0 \times 125^0 = (5^0)^6$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the equation and check if they are equal.

Consider the left side (LHS) of the equation: $5^0 \times 25^0 \times 125^0$.

We use the property that any non-zero number raised to the power of 0 is equal to 1 ($a^0 = 1$ for $a \neq 0$).

Since 5, 25, and 125 are all non-zero numbers, we have:

$5^0 = 1$

$25^0 = 1$

$125^0 = 1$

Substitute these values into the LHS:

LHS $= 1 \times 1 \times 1 = 1$.

Now, consider the right side (RHS) of the equation: $(5^0)^6$.

First, evaluate the term inside the parentheses, $5^0$. As established, $5^0 = 1$.

So, the RHS becomes $(1)^6$.

Any power of 1 is equal to 1:

$(1)^6 = 1 \times 1 \times 1 \times 1 \times 1 \times 1 = 1$.

So, RHS $= 1$.

Comparing the left side and the right side:

LHS $= 1$

RHS $= 1$

Since $1 = 1$, the equality holds true.

Therefore, the given statement "$5^0 \times 25^0 \times 125^0 = (5^0)^6$" is correct.

The statement is True.

Given:

The statement is $876543 = 8 \times 10^5 + 7 \times 10^4 + 6 \times 10^3 + 5 \times 10^2 + 4 \times 10^1 + 3 \times 10^0$.

To Determine:

Whether the given statement is True or False.

Solution:

The right side of the equation is the expanded form of a number using powers of 10. We need to evaluate the sum on the right side and check if it is equal to the number on the left side.

Let's evaluate each term on the right side:

$10^0 = 1$

$10^1 = 10$

$10^2 = 10 \times 10 = 100$

$10^3 = 10 \times 10 \times 10 = 1000$

$10^4 = 10 \times 10 \times 10 \times 10 = 10000$

$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100000$

Now, substitute these values back into the right side of the equation:

RHS $= 8 \times 100000 + 7 \times 10000 + 6 \times 1000 + 5 \times 100 + 4 \times 10 + 3 \times 1$

Perform the multiplications:

RHS $= 800000 + 70000 + 6000 + 500 + 40 + 3$

Now, perform the addition:

$\begin{array}{cc} & 800000 \\ + & \phantom{0}70000 \\ + & \phantom{00}6000 \\ + & \phantom{000}500 \\ + & \phantom{0000}40 \\ + & \phantom{00000}3 \\ \hline & 876543 \\ \hline \end{array}$

The sum on the right side is 876543.

The left side of the equation is 876543.

Comparing the left side and the right side, we see that they are equal:

$876543 = 876543$

This is the correct expanded form of the number 876543 based on its place values.

Therefore, the given statement is correct.

The statement is True.

Given:

The statement is $600060 = 6 \times 10^5 + 6 \times 10^2$.

To Determine:

Whether the given statement is True or False.

Solution:

To verify the truthfulness of the statement, we need to evaluate the expression on the right side of the equation and compare it with the number on the left side.

The right side (RHS) is $6 \times 10^5 + 6 \times 10^2$.

Let's evaluate the terms involving powers of 10:

$10^5 = 100000$

$10^2 = 100$

Substitute these values back into the expression for the RHS:

RHS $= 6 \times 100000 + 6 \times 100$

Perform the multiplications:

$6 \times 100000 = 600000$

$6 \times 100 = 600$

Now, perform the addition:

RHS $= 600000 + 600 = 600600$

The left side (LHS) of the equation is 600060.

Comparing the LHS and RHS:

$600060 = 600600$

This equality is false, as 600060 is not equal to 600600.

The number 600060 in expanded form is $6 \times 10^5 + 0 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 6 \times 10^1 + 0 \times 10^0$, which simplifies to $6 \times 10^5 + 6 \times 10^1$.

Therefore, the given statement is incorrect.

The statement is False.

Given:

The statement is $4 \times 10^5 + 3 \times 10^4 + 2 \times 10^3 + 1 \times 10^0 = 432010$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the expression on the left side of the equation and compare it with the number on the right side.

The left side (LHS) is $4 \times 10^5 + 3 \times 10^4 + 2 \times 10^3 + 1 \times 10^0$.

Let's evaluate each term involving powers of 10:

$10^5 = 100000$

$10^4 = 10000$

$10^3 = 1000$

$10^0 = 1$ (Any non-zero number raised to the power of 0 is 1).

Now, substitute these values back into the expression for the LHS:

LHS $= 4 \times 100000 + 3 \times 10000 + 2 \times 1000 + 1 \times 1$

Perform the multiplications:

$4 \times 100000 = 400000$

$3 \times 10000 = 30000$

$2 \times 1000 = 2000$

$1 \times 1 = 1$

Now, perform the addition of these results:

LHS $= 400000 + 30000 + 2000 + 1$

$\begin{array}{cc} & 400000 \\ + & \phantom{0}30000 \\ + & \phantom{00}2000 \\ + & \phantom{00000}1 \\ \hline & 432001 \\ \hline \end{array}$

The left side of the equation evaluates to 432001.

The right side of the equation is 432010.

Comparing the LHS and RHS:

$432001 = 432010$

This equality is false, as 432001 is not equal to 432010.

The given statement is incorrect.

The statement is False.

Given:

The statement is $8 \times 10^6 + 2 \times 10^4 + 5 \times 10^2 + 9 \times 10^0 = 8020509$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate the expression on the left side of the equation and compare the result with the number on the right side.

The left side (LHS) is $8 \times 10^6 + 2 \times 10^4 + 5 \times 10^2 + 9 \times 10^0$.

Let's evaluate each term involving powers of 10:

$10^6 = 1,000,000$

$10^4 = 10,000$

$10^2 = 100$

$10^0 = 1$ (Any non-zero number raised to the power of 0 is 1).

Now, substitute these values back into the expression for the LHS:

LHS $= 8 \times 1,000,000 + 2 \times 10,000 + 5 \times 100 + 9 \times 1$

Perform the multiplications:

$8 \times 1,000,000 = 8,000,000$

$2 \times 10,000 = 20,000$

$5 \times 100 = 500$

$9 \times 1 = 9$

Now, perform the addition of these results:

LHS $= 8,000,000 + 20,000 + 500 + 9$

$\begin{array}{cc} & 8000000 \\ + & \phantom{0}20000 \\ + & \phantom{000}500 \\ + & \phantom{00000}9 \\ \hline & 8020509 \\ \hline \end{array}$

The left side of the equation evaluates to 8020509.

The right side (RHS) of the equation is 8020509.

Comparing the LHS and RHS:

$8020509 = 8020509$

This equality is true. The expression on the left side is the expanded form of the number 8020509, considering the place values corresponding to the powers of 10 present.

Therefore, the given statement is correct.

The statement is True.

Given:

The statement is $4^0 + 5^0 + 6^0 = (4 + 5 + 6)^0$.

To Determine:

Whether the given statement is True or False.

Solution:

We need to evaluate both sides of the equation and check if they are equal.

Consider the left side (LHS) of the equation: $4^0 + 5^0 + 6^0$.

We use the property of exponents which states that any non-zero number raised to the power of 0 is equal to 1.

$a^0 = 1$, for $a \neq 0$.

Since 4, 5, and 6 are all non-zero numbers, we have:

$4^0 = 1$

$5^0 = 1$

$6^0 = 1$

Substitute these values into the LHS expression:

LHS $= 1 + 1 + 1$

LHS $= 3$

Now, consider the right side (RHS) of the equation: $(4 + 5 + 6)^0$.

First, evaluate the sum inside the parentheses:

$4 + 5 + 6 = 15$

So, the RHS becomes $(15)^0$.

Using the same property $a^0 = 1$ for $a \neq 0$, and since 15 is a non-zero number:

$15^0 = 1$

So, RHS $= 1$.

Comparing the left side and the right side of the original statement:

$3 = 1$

This equality is false.

Therefore, the given statement "$4^0 + 5^0 + 6^0 = (4 + 5 + 6)^0$" is incorrect.

The statement is False.

First, we evaluate the value of each given term:

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

$33 = 33$

$2^3 \times 2 = 2^{3+1} = 2^4 = 2 \times 2 \times 2 \times 2 = 16$

$(3^3)^2 = 3^{3 \times 2} = 3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$

$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$

$4^0 = 1$ (Any non-zero number raised to the power of 0 is 1)

$2^3 \times 3^1 = 8 \times 3 = 24$

The values of the given terms are: $32, 33, 16, 729, 243, 1, 24$.

Arranging these values in ascending order, we get:

$1, 16, 24, 32, 33, 243, 729$.

Now, we replace these values with their original expressions:

$1$ corresponds to $4^0$

$16$ corresponds to $2^3 \times 2$

$24$ corresponds to $2^3 \times 3^1$

$32$ corresponds to $2^5$

$33$ corresponds to $33$

$243$ corresponds to $3^5$

$729$ corresponds to $(3^3)^2$

Therefore, the terms arranged in ascending order are:

$4^0$, $2^3 \times 2$, $2^3 \times 3^1$, $2^5$, $33$, $3^5$, $(3^3)^2$.

First, we evaluate the value of each given term:

$2^{2+3} = 2^5 = 32$

$(2^2)^3 = 2^{2 \times 3} = 2^6 = 64$

$2 \times 2^2 = 2^{1+2} = 2^3 = 8$

$\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$

$3^2 \times 3^0 = 3^{2+0} = 3^2 = 9$

$2^3 \times 5^2 = 8 \times 25 = 200$

The values of the given terms are: $32, 64, 8, 27, 9, 200$.

Arranging these values in descending order, we get:

$200, 64, 32, 27, 9, 8$.

Now, we replace these values with their original expressions:

$200$ corresponds to $2^3 \times 5^2$

$64$ corresponds to $(2^2)^3$

$32$ corresponds to $2^{2+3}$

$27$ corresponds to $\frac{3^5}{3^2}$

$9$ corresponds to $3^2 \times 3^0$

$8$ corresponds to $2 \times 2^2$

Therefore, the terms arranged in descending order are:

$2^3 \times 5^2$, $(2^2)^3$, $2^{2+3}$, $\frac{3^5}{3^2}$, $3^2 \times 3^0$, $2 \times 2^2$.

Let the number by which $(-4)^5$ should be divided be $x$.

According to the question, we have:

$(-4)^5 \div x = (-4)^3$

We can write this equation as:

$\frac{(-4)^5}{x} = (-4)^3$

To find $x$, we can rearrange the equation:

$x = \frac{(-4)^5}{(-4)^3}$

Using the property of exponents, $\frac{a^m}{a^n} = a^{m-n}$, we have:

$x = (-4)^{5-3}$

$x = (-4)^2$

Now, we calculate the value of $(-4)^2$:

$(-4)^2 = (-4) \times (-4) = 16$

So, the number is 16.

Thus, $(-4)^5$ should be divided by 16 so that the quotient is equal to $(-4)^3$.

Given the equation:

$\left( \frac{2}{9} \right)^{3}$ × $\left( \frac{2}{9} \right)^{6}$ = $\left( \frac{2}{9} \right)^{2m \;-\; 1}$

Using the property of exponents $a^m \times a^n = a^{m+n}$ on the left side of the equation, we combine the terms with the same base:

$\left( \frac{2}{9} \right)^{3+6} = \left( \frac{2}{9} \right)^{2m \;-\; 1}$

$\left( \frac{2}{9} \right)^{9} = \left( \frac{2}{9} \right)^{2m \;-\; 1}$

Since the bases on both sides of the equation are equal and non-zero ($2/9 \neq 0$) and not equal to 1 or -1, their exponents must be equal.

Equating the exponents, we get:

$9 = 2m - 1$

Now, we solve this linear equation for $m$:

Add 1 to both sides of the equation:

$9 + 1 = 2m - 1 + 1$

$10 = 2m$

Divide both sides by 2:

$\frac{10}{2} = \frac{2m}{2}$

$m = 5$

Thus, the value of $m$ is 5.

Given the expression for $\frac{p}{q}$:

$\frac{p}{q} = \left( \frac{3}{2} \right)^{2} \div \left( \frac{9}{4} \right)^{0}$

First, we evaluate the terms on the right side.

Calculate $\left( \frac{3}{2} \right)^{2}$:

$\left( \frac{3}{2} \right)^{2} = \frac{3^2}{2^2} = \frac{9}{4}$

Calculate $\left( \frac{9}{4} \right)^{0}$. Using the property that any non-zero number raised to the power of 0 is 1, we have:

$\left( \frac{9}{4} \right)^{0} = 1$

Now substitute these values back into the expression for $\frac{p}{q}$:

$\frac{p}{q} = \frac{9}{4} \div 1$

$\frac{p}{q} = \frac{9}{4}$

Next, we need to find the value of $\left( \frac{p}{q} \right)^{3}$.

Substitute the value of $\frac{p}{q} = \frac{9}{4}$:

$\left( \frac{p}{q} \right)^{3} = \left( \frac{9}{4} \right)^{3}$

Calculate $\left( \frac{9}{4} \right)^{3}$:

$\left( \frac{9}{4} \right)^{3} = \frac{9^3}{4^3}$

$9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$

So,

$\left( \frac{p}{q} \right)^{3} = \frac{729}{64}$

Thus, the value of $\left( \frac{p}{q} \right)^{3}$ is $\frac{729}{64}$.

Let the given rational number be $N$.

$N = \left( \frac{1}{2} \right)^{2} \div \left( \frac{2}{3} \right)^{3}$

First, we evaluate the powers:

$\left( \frac{1}{2} \right)^{2} = \frac{1^2}{2^2} = \frac{1}{4}$

$\left( \frac{2}{3} \right)^{3} = \frac{2^3}{3^3} = \frac{8}{27}$

Now, substitute these values back into the expression for $N$:

$N = \frac{1}{4} \div \frac{8}{27}$

To divide by a fraction, we multiply by its reciprocal:

$N = \frac{1}{4} \times \frac{27}{8}$

Multiply the numerators and the denominators:

$N = \frac{1 \times 27}{4 \times 8} = \frac{27}{32}$

We need to find the reciprocal of this rational number $N = \frac{27}{32}$.

The reciprocal of a rational number $\frac{a}{b}$ is $\frac{b}{a}$, provided $a \neq 0$ and $b \neq 0$.

The reciprocal of $\frac{27}{32}$ is $\frac{32}{27}$.

Thus, the reciprocal of the given rational number is $\frac{32}{27}$.

We need to find the value of each expression.

Recall that for any non-zero number $a$, $a^0 = 1$.

(a) Evaluate $7^0$:

$7^0 = 1$

(b) Evaluate $7^7 \div 7^7$:

Using the property $a^m \div a^n = a^{m-n}$:

$7^7 \div 7^7 = 7^{7-7} = 7^0 = 1$

Alternatively, any non-zero number divided by itself is 1.

$7^7 \div 7^7 = \frac{7^7}{7^7} = 1$

(c) Evaluate $(-7)^{2 \times 7 – 6 – 8}$:

First, evaluate the exponent:

$2 \times 7 - 6 - 8 = 14 - 6 - 8 = 8 - 8 = 0$

So the expression is $(-7)^0$.

$(-7)^0 = 1$

(d) Evaluate $(2^0 + 3^0 + 4^0) (4^0 – 3^0 – 2^0)$:

Evaluate each term with exponent 0:

$2^0 = 1$

$3^0 = 1$

$4^0 = 1$

Substitute these values into the expression:

$(1 + 1 + 1) (1 - 1 - 1)$

$(3)(-1)$

$3 \times (-1) = -3$

(e) Evaluate $2 \times 3 \times 4 \div 2^0 \times 3^0 \times 4^0$:

Evaluate the terms with exponent 0:

$2^0 = 1$

$3^0 = 1$

$4^0 = 1$

Substitute these values into the expression and follow the order of operations (division and multiplication from left to right):

$2 \times 3 \times 4 \div 1 \times 1 \times 1$

$6 \times 4 \div 1 \times 1 \times 1$

$24 \div 1 \times 1 \times 1$

$24 \times 1 \times 1$

$24$

(f) Evaluate $(8^0 – 2^0) \times (8^0 + 2^0)$:

Evaluate the terms with exponent 0:

$8^0 = 1$

$2^0 = 1$

Substitute these values into the expression:

$(1 - 1) \times (1 + 1)$

$(0) \times (2)$

$0 \times 2 = 0$

Given the equation:

$2^{n – 5} \times 6^{2n – 4} = \frac{1}{12^{4} \;\times\; 2}$

Let's simplify the left side (LHS) by expressing the base 6 in terms of its prime factors:

$6 = 2 \times 3$

So, the LHS is:

$2^{n – 5} \times (2 \times 3)^{2n – 4}$

Using the exponent property $(ab)^m = a^m b^m$:

$= 2^{n – 5} \times 2^{2n – 4} \times 3^{2n – 4}$

Using the exponent property $a^m \times a^n = a^{m+n}$ for the base 2 terms:

$= 2^{(n – 5) + (2n – 4)} \times 3^{2n – 4}$

$= 2^{n – 5 + 2n – 4} \times 3^{2n – 4}$

$= 2^{3n – 9} \times 3^{2n – 4}$

Now, let's simplify the right side (RHS) by expressing the base 12 in terms of its prime factors:

$12 = 2^2 \times 3$

So, the RHS is:

$\frac{1}{(2^2 \times 3)^{4} \times 2}$

Using the exponent properties $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$:

$= \frac{1}{(2^2)^4 \times 3^4 \times 2^1}$

$= \frac{1}{2^8 \times 3^4 \times 2^1}$

Using the exponent property $a^m \times a^n = a^{m+n}$ in the denominator:

$= \frac{1}{2^{8+1} \times 3^4} = \frac{1}{2^9 \times 3^4}$

Rewrite using negative exponents $1/a^m = a^{-m}$ and $1/b^n = b^{-n}$:

$= 2^{-9} \times 3^{-4}$

Equating the simplified LHS and RHS:

$2^{3n – 9} \times 3^{2n – 4} = 2^{-9} \times 3^{-4}$

Since the bases (2 and 3) are prime numbers greater than 1, and the expressions are equal, the exponents of the corresponding bases on both sides must be equal.

Equating the exponents of base 2:

$3n - 9 = -9$

Equating the exponents of base 3:

$2n - 4 = -4$

Let's solve the equation obtained from equating the exponents of base 2:

$3n - 9 = -9$

Add 9 to both sides of the equation:

$3n - 9 + 9 = -9 + 9$

$3n = 0$

Divide both sides by 3:

$n = \frac{0}{3}$

$n = 0$

Let's solve the equation obtained from equating the exponents of base 3 (this will verify our value of $n$):

$2n - 4 = -4$

Add 4 to both sides of the equation:

$2n - 4 + 4 = -4 + 4$

$2n = 0$

Divide both sides by 2:

$n = \frac{0}{2}$

$n = 0$

Both equations yield the same value, $n = 0$. The problem states that $n$ is an integer, which 0 is.

Thus, the value of $n$ is 0.

To express a number from standard form ($a \times 10^n$) to usual form:

If the exponent $n$ is positive, move the decimal point $n$ places to the right.

If the exponent $n$ is negative, move the decimal point $|n|$ places to the left.

(a) Express $8.01 \times 10^7$ in usual form:

The exponent is $7$, which is positive.

We move the decimal point in $8.01$ seven places to the right.

$8.01 \times 10^7 = 80,100,000$

(b) Express $1.75 \times 10^{-3}$ in usual form:

The exponent is $-3$, which is negative.

We move the decimal point in $1.75$ three places to the left.

$1.75 \times 10^{-3} = 0.00175$

(a) To find the value of $2^5$, we multiply 2 by itself 5 times:

$2^5 = 2 \times 2 \times 2 \times 2 \times 2$

$2 \times 2 = 4$

$4 \times 2 = 8$

$8 \times 2 = 16$

$16 \times 2 = 32$

So, the value of $2^5$ is 32.

(b) To find the value of $(-3^5)$, the exponent applies only to the base 3. The negative sign is outside the power.

First, calculate $3^5$:

$3^5 = 3 \times 3 \times 3 \times 3 \times 3$

$3 \times 3 = 9$

$9 \times 3 = 27$

$27 \times 3 = 81$

$81 \times 3 = 243$

So, $3^5 = 243$.

Now, apply the negative sign:

$-3^5 = -(243) = -243$

So, the value of $(-3^5)$ is -243.

(c) To find the value of $-(-4)^4$, the exponent 4 applies to the base $(-4)$. The negative sign outside is applied after calculating the power.

First, calculate $(-4)^4$:

$(-4)^4 = (-4) \times (-4) \times (-4) \times (-4)$

$(-4) \times (-4) = 16$

$16 \times (-4) = -64$

$(-64) \times (-4) = 256$

So, $(-4)^4 = 256$.

Now, apply the negative sign outside:

$-(-4)^4 = -(256) = -256$

So, the value of $-(-4)^4$ is -256.

To express a product in exponential form, we identify the base being multiplied repeatedly and count the number of times it is multiplied. The number of times the base is multiplied becomes the exponent.

(a) Given expression: $3 \times 3 \times 3 \times a \times a \times a \times a$

The base 3 is multiplied 3 times. Its exponential form is $3^3$.

The base $a$ is multiplied 4 times. Its exponential form is $a^4$.

Combining these, the exponential form of the given expression is:

$3^3 a^4$

(b) Given expression: $a \times a \times b \times b \times b \times c \times c \times c \times c$

The base $a$ is multiplied 2 times. Its exponential form is $a^2$.

The base $b$ is multiplied 3 times. Its exponential form is $b^3$.

The base $c$ is multiplied 4 times. Its exponential form is $c^4$.

Combining these, the exponential form of the given expression is:

$a^2 b^3 c^4$

(c) Given expression: $s \times s \times t \times t \times s \times s \times t$

Let's count the occurrences of each base:

The base $s$ appears 4 times ($s \times s \times s \times s$). Its exponential form is $s^4$.

The base $t$ appears 3 times ($t \times t \times t$). Its exponential form is $t^3$.

Combining these, the exponential form of the given expression is:

$s^4 t^3$

Let the number of times 30 must be added together be $x$.

Adding 30 together $x$ times is equivalent to multiplying 30 by $x$.

So, the problem can be expressed as the equation:

$x \times 30 = 30^7$

To find the value of $x$, we need to isolate it. Divide both sides of the equation by 30:

$x = \frac{30^7}{30}$

We can write $30$ as $30^1$. Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, where $a=30$, $m=7$, and $n=1$:

$x = 30^{7-1}$

$x = 30^6$

Thus, 30 must be added together $30^6$ times to get a sum equal to $30^7$.

The number of times 30 must be added is $30^6$.

To express a number in exponential notation, we find its prime factorization and write it as a product of powers of its prime factors.

(a) Express 1024 in exponential notation:

We find the prime factorization of 1024:

So, $1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{10}$.

The exponential notation for 1024 is $2^{10}$.

(b) Express 1029 in exponential notation:

We find the prime factorization of 1029:

So, $1029 = 3 \times 7 \times 7 \times 7 = 3^1 \times 7^3$.

The exponential notation for 1029 is $3^1 \times 7^3$ or $3 \times 7^3$.

(c) Express $\frac{144}{875}$ in exponential notation:

We find the prime factorization of the numerator, 144:

So, $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2$.

Next, we find the prime factorization of the denominator, 875:

So, $875 = 5 \times 5 \times 5 \times 7 = 5^3 \times 7^1$.

Now, we express the fraction using these exponential forms:

$\frac{144}{875} = \frac{2^4 \times 3^2}{5^3 \times 7^1}$

The exponential notation for $\frac{144}{875}$ is $\frac{2^4 \times 3^2}{5^3 \times 7^1}$ or $\frac{2^4 \times 3^2}{5^3 \times 7}$.

(a) To identify the greater number between $2^6$ and $6^2$, we calculate their values:

$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$

$6^2 = 6 \times 6 = 36$

Comparing the values, we have $64 > 36$.

Therefore, $2^6$ is the greater number.

(b) To identify the greater number between $2^9$ and $9^2$, we calculate their values:

$2^9 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 512$

$9^2 = 9 \times 9 = 81$

Comparing the values, we have $512 > 81$.

Therefore, $2^9$ is the greater number.

(c) To identify the greater number between $7.9 \times 10^4$ and $5.28 \times 10^5$, we compare their exponents of 10.

The first number is $7.9 \times 10^4$, where the exponent is 4.

The second number is $5.28 \times 10^5$, where the exponent is 5.

Since $5 > 4$, the number with the exponent 5 is greater.

Alternatively, we can express both numbers in usual form or with the same power of 10.

In usual form:

$7.9 \times 10^4 = 79,000$

$5.28 \times 10^5 = 528,000$

Comparing the values, we have $528,000 > 79,000$.

With the same power of 10 (e.g., $10^4$):

$7.9 \times 10^4$

$5.28 \times 10^5 = 5.28 \times 10^1 \times 10^4 = 52.8 \times 10^4$

Comparing $7.9 \times 10^4$ and $52.8 \times 10^4$, since $52.8 > 7.9$, the second number is greater.

Therefore, $5.28 \times 10^5$ is the greater number.

To express a number as a product of powers of its prime factors, we perform prime factorization.

(a) Express 9000 as a product of powers of its prime factors:

We find the prime factorization of 9000:

So, $9000 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5$.

In exponential notation, this is $2^3 \times 3^2 \times 5^3$.

The product of powers of prime factors for 9000 is $2^3 \times 3^2 \times 5^3$.

(b) Express 2025 as a product of powers of its prime factors:

We find the prime factorization of 2025:

So, $2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5$.

In exponential notation, this is $3^4 \times 5^2$.

The product of powers of prime factors for 2025 is $3^4 \times 5^2$.

(c) Express 800 as a product of powers of its prime factors:

We find the prime factorization of 800:

So, $800 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5$.

In exponential notation, this is $2^5 \times 5^2$.

The product of powers of prime factors for 800 is $2^5 \times 5^2$.

(a) Express $2^3 \times 3^3$ in single exponential form:

Here, the exponents are the same (3), but the bases are different (2 and 3).

Using the property $(a \times b)^m = a^m \times b^m$, we can write:

$2^3 \times 3^3 = (2 \times 3)^3 = 6^3$

The single exponential form is $6^3$.

(b) Express $2^4 \times 4^2$ in single exponential form:

Here, the bases are different (2 and 4) and the exponents are different (4 and 2).

We can express the base 4 as a power of 2: $4 = 2^2$.

Substitute $4^2 = (2^2)^2$ into the expression:

$2^4 \times (2^2)^2$

Using the property $(a^m)^n = a^{m \times n}$:

$= 2^4 \times 2^{2 \times 2} = 2^4 \times 2^4$

Now the bases are the same (2). Using the property $a^m \times a^n = a^{m+n}$:

$= 2^{4+4} = 2^8$

Alternatively, we could express both terms with the same exponent:

$2^4 \times 4^2 = (2^2)^2 \times 4^2 = 4^2 \times 4^2$

Using the property $a^m \times a^n = a^{m+n}$:

$= 4^{2+2} = 4^4$

Note that $2^8 = 256$ and $4^4 = (2^2)^4 = 2^8 = 256$. Both forms are valid single exponential forms.

The single exponential form is $2^8$ or $4^4$.

(c) Express $5^2 \times 7^2$ in single exponential form:

Here, the exponents are the same (2), but the bases are different (5 and 7).

Using the property $(a \times b)^m = a^m \times b^m$, we can write:

$5^2 \times 7^2 = (5 \times 7)^2 = 35^2$

The single exponential form is $35^2$.

(d) Express $(-5)^5 \times (-5)$ in single exponential form:

Here, the bases are the same $(-5)$. Note that $(-5)$ can be written as $(-5)^1$.

Using the property $a^m \times a^n = a^{m+n}$:

$(-5)^5 \times (-5)^1 = (-5)^{5+1} = (-5)^6$

The single exponential form is $(-5)^6$.

(e) Express $(-3)^3 \times (-10)^3$ in single exponential form:

Here, the exponents are the same (3), but the bases are different (-3 and -10).

Using the property $(a \times b)^m = a^m \times b^m$, we can write:

$(-3)^3 \times (-10)^3 = ((-3) \times (-10))^3$

Calculate the product of the bases: $(-3) \times (-10) = 30$.

$= 30^3$

The single exponential form is $30^3$.

(f) Express $(-11)^2 \times (-2)^2$ in single exponential form:

Here, the exponents are the same (2), but the bases are different (-11 and -2).

Using the property $(a \times b)^m = a^m \times b^m$, we can write:

$(-11)^2 \times (-2)^2 = ((-11) \times (-2))^2$

Calculate the product of the bases: $(-11) \times (-2) = 22$.

$= 22^2$

The single exponential form is $22^2$.

To express a number in standard form (scientific notation), we write it as a number between 1 and 10 (inclusive of 1 but exclusive of 10), multiplied by a power of 10. The format is $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.

(a) Express 76,47,000 in standard form:

The number is 7,647,000.

The decimal point is initially at the end of the number.

We move the decimal point to the left until it is after the first non-zero digit (7).

The number of places moved is 6 (from the end to after 7).

Since the original number is large, the exponent of 10 is positive.

$7,647,000 = 7.647 \times 10^6$

The standard form is $7.647 \times 10^6$.

(b) Express 8,19,00,000 in standard form:

The number is 81,900,000.

The decimal point is initially at the end of the number.

We move the decimal point to the left until it is after the first non-zero digit (8).

The number of places moved is 7 (from the end to after 8).

Since the original number is large, the exponent of 10 is positive.

$81,900,000 = 8.19 \times 10^7$

The standard form is $8.19 \times 10^7$.

(c) Express 5, 83,00,00,00,000 in standard form:

The number is 583,000,000,000.

The decimal point is initially at the end of the number.

We move the decimal point to the left until it is after the first non-zero digit (5).

The number of places moved is 11 (from the end to after 5).

Since the original number is large, the exponent of 10 is positive.

$583,000,000,000 = 5.83 \times 10^{11}$

The standard form is $5.83 \times 10^{11}$.

(d) Express 24 billion in standard form:

One billion is $1,000,000,000$, which is $10^9$.

So, 24 billion $= 24 \times 10^9$.

To express 24 in standard form, we write it as a number between 1 and 10 multiplied by a power of 10.

$24 = 2.4 \times 10^1$.

Substitute this back into the expression:

$24 \times 10^9 = (2.4 \times 10^1) \times 10^9$

Using the exponent property $10^m \times 10^n = 10^{m+n}$:

$= 2.4 \times 10^{1+9} = 2.4 \times 10^{10}$

The standard form is $2.4 \times 10^{10}$.

Given:

Speed of light in vacuum ($v$) = $3 \times 10^8$ m/s

Time taken for sunlight to reach Earth ($t$) = 8 minutes

To Find:

Distance of Sun from Earth in standard form.

Solution:

First, we need to convert the time from minutes to seconds, since the speed is given in meters per second.

1 minute = 60 seconds

Time in seconds ($t$) = 8 minutes $\times$ 60 seconds/minute

$t = 480$ seconds

The relationship between distance, speed, and time is given by:

Distance = Speed $\times$ Time

$d = v \times t$

Substitute the given values into the formula:

$d = (3 \times 10^8 \text{ m/s}) \times (480 \text{ s})$

$d = 3 \times 480 \times 10^8$ meters

$d = 1440 \times 10^8$ meters

Now, we need to express this distance in standard form ($a \times 10^n$, where $1 \leq |a| < 10$).

We need to write 1440 in standard form. The decimal point is after the last zero.

$1440 = 1.440 \times 10^3$

So, the distance is:

$d = (1.44 \times 10^3) \times 10^8$ meters

Using the property of exponents $10^m \times 10^n = 10^{m+n}$:

$d = 1.44 \times 10^{3+8}$ meters

$d = 1.44 \times 10^{11}$ meters

The distance of the Sun from Earth in standard form is $1.44 \times 10^{11}$ meters.

We use the properties of exponents to simplify each expression:

• $a^m \times a^n = a^{m+n}$ (Product of powers)
• $a^m \div a^n = a^{m-n}$ (Quotient of powers)
• $(a^m)^n = a^{mn}$ (Power of a power)
• $a^0 = 1$ (Zero exponent)

(a) Simplify $\left[ \left( \frac{3}{7} \right)^{4} \times \left( \frac{3}{7} \right)^{5} \right] \div \left( \frac{3}{7} \right)^{7}$

First, simplify the expression inside the square brackets using the product of powers rule:

$\left( \frac{3}{7} \right)^{4} \times \left( \frac{3}{7} \right)^{5} = \left( \frac{3}{7} \right)^{4+5} = \left( \frac{3}{7} \right)^{9}$

Now, substitute this back into the original expression:

$\left( \frac{3}{7} \right)^{9} \div \left( \frac{3}{7} \right)^{7}$

Using the quotient of powers rule:

$\left( \frac{3}{7} \right)^{9-7} = \left( \frac{3}{7} \right)^{2}$

The exponential form is $\left( \frac{3}{7} \right)^{2}$.

(b) Simplify $\left[ \left( \frac{7}{11} \right)^{5} \div \left( \frac{7}{11} \right)^{2} \right] \times \left( \frac{7}{11} \right)^{2}$

First, simplify the expression inside the square brackets using the quotient of powers rule:

$\left( \frac{7}{11} \right)^{5} \div \left( \frac{7}{11} \right)^{2} = \left( \frac{7}{11} \right)^{5-2} = \left( \frac{7}{11} \right)^{3}$

Now, substitute this back into the original expression:

$\left( \frac{7}{11} \right)^{3} \times \left( \frac{7}{11} \right)^{2}$

Using the product of powers rule:

$\left( \frac{7}{11} \right)^{3+2} = \left( \frac{7}{11} \right)^{5}$

The exponential form is $\left( \frac{7}{11} \right)^{5}$.

(c) Simplify $(3^7 \div 3^5)^4$

First, simplify the expression inside the parentheses using the quotient of powers rule:

$3^7 \div 3^5 = 3^{7-5} = 3^2$

Now, substitute this back into the original expression:

$(3^2)^4$

Using the power of a power rule:

$3^{2 \times 4} = 3^8$

The exponential form is $3^8$.

(d) Simplify $\left( \frac{a^{6}}{a^{4}} \right)$ × a⁵ × a⁰

First, simplify the expression inside the parentheses using the quotient of powers rule:

$\frac{a^6}{a^4} = a^{6-4} = a^2$

Now, substitute this back into the original expression:

$a^2 \times a^5 \times a^0$

We know that $a^0 = 1$ (assuming $a \neq 0$). Using the product of powers rule:

$a^2 \times a^5 \times 1 = a^{2+5} \times 1 = a^7 \times 1 = a^7$

The exponential form is $a^7$ (assuming $a \neq 0$).

(e) Simplify $\left[ \left( \frac{3}{5} \right)^{3} \times \left( \frac{3}{5} \right)^{8} \right] \div \left[ \left( \frac{3}{5} \right)^{2} \times \left( \frac{3}{5} \right)^{4} \right]$

Simplify the first part in the square brackets using the product of powers rule:

$\left( \frac{3}{5} \right)^{3} \times \left( \frac{3}{5} \right)^{8} = \left( \frac{3}{5} \right)^{3+8} = \left( \frac{3}{5} \right)^{11}$

Simplify the second part in the square brackets using the product of powers rule:

$\left( \frac{3}{5} \right)^{2} \times \left( \frac{3}{5} \right)^{4} = \left( \frac{3}{5} \right)^{2+4} = \left( \frac{3}{5} \right)^{6}$

Now, substitute these results back into the original expression:

$\left( \frac{3}{5} \right)^{11} \div \left( \frac{3}{5} \right)^{6}$

Using the quotient of powers rule:

$\left( \frac{3}{5} \right)^{11-6} = \left( \frac{3}{5} \right)^{5}$

The exponential form is $\left( \frac{3}{5} \right)^{5}$.

(f) Simplify $(5^{15} \div 5^{10}) \times 5^5$

First, simplify the expression inside the parentheses using the quotient of powers rule:

$5^{15} \div 5^{10} = 5^{15-10} = 5^5$

Now, substitute this back into the original expression:

$5^5 \times 5^5$

Using the product of powers rule:

$5^{5+5} = 5^{10}$

The exponential form is $5^{10}$.

(a) Evaluate $\frac{7^{8} \;\times\; a^{10}b^{7}c^{12}}{7^{6} \;\times\; a^{8}b^{4}c^{12}}$:

We can use the quotient of powers rule $\frac{x^m}{x^n} = x^{m-n}$ for each base:

$\frac{7^8}{7^6} = 7^{8-6} = 7^2$

$\frac{a^{10}}{a^8} = a^{10-8} = a^2$

$\frac{b^7}{b^4} = b^{7-4} = b^3$

$\frac{c^{12}}{c^{12}} = c^{12-12} = c^0 = 1$ (assuming $c \neq 0$)

Combine the simplified terms:

$\frac{7^{8} \;\times\; a^{10}b^{7}c^{12}}{7^{6} \;\times\; a^{8}b^{4}c^{12}} = 7^2 \times a^2 \times b^3 \times 1 = 7^2 a^2 b^3$

Calculate the value of $7^2 = 49$.

The value is $49 a^2 b^3$ (assuming $a, b, c \neq 0$).

(b) Evaluate $\frac{5^{4} \;\times\; 7^{4} \;\times\; 2^{7}}{8 \;\times\; 49 \;\times\; 5^{3}}$:

Express the composite bases in terms of prime factors:

$8 = 2^3$

$49 = 7^2$

Substitute these into the expression:

$\frac{5^{4} \;\times\; 7^{4} \;\times\; 2^{7}}{2^{3} \;\times\; 7^{2} \;\times\; 5^{3}}$

Rearrange the terms to group by base:

$\left( \frac{5^4}{5^3} \right) \times \left( \frac{7^4}{7^2} \right) \times \left( \frac{2^7}{2^3} \right)$

Use the quotient of powers rule:

$5^{4-3} \times 7^{4-2} \times 2^{7-3}$

$= 5^1 \times 7^2 \times 2^4$

$= 5 \times 49 \times 16$

Calculate the product:

$5 \times 49 = 245$

$245 \times 16$

$\begin{array}{cc}& & 2 & 4 & 5 \\ \times & & & 1 & 6 \\ \hline && 14 & 7 & 0 \\ & 2 & 4 & 5 & \times \\ \hline 3 & 9 & 2 & 0 \\ \hline \end{array}$

The value is 3920.

(c) Evaluate $\frac{125 \;\times\; 5^{2} \;\times\; a^{7}}{10^{3} \;\times\; a^{4}}$:

Express the composite bases in terms of prime factors:

$125 = 5^3$

$10 = 2 \times 5$

Substitute these into the expression:

$\frac{5^3 \;\times\; 5^{2} \;\times\; a^{7}}{(2 \times 5)^{3} \;\times\; a^{4}}$

Use the product of powers rule in the numerator and the power of a product rule in the denominator:

$\frac{5^{3+2} \;\times\; a^{7}}{2^3 \;\times\; 5^3 \;\times\; a^{4}} = \frac{5^5 \;\times\; a^{7}}{2^3 \;\times\; 5^3 \;\times\; a^{4}}$

Rearrange and use the quotient of powers rule:

$\frac{5^5}{5^3} \times \frac{a^7}{a^4} \times \frac{1}{2^3}$

$= 5^{5-3} \times a^{7-4} \times \frac{1}{2^3}$

$= 5^2 \times a^3 \times \frac{1}{2^3}$

Calculate the values of the numbers:

$5^2 = 25$

$2^3 = 8$

The value is $\frac{25 a^3}{8}$ (assuming $a \neq 0$).

(d) Evaluate $\frac{3^{4} \;\times\; 12^{3} \;\times\; 36}{2^{5} \;\times\; 6^{3}}$:

Express the composite bases in terms of prime factors:

$12 = 2^2 \times 3$

$36 = 6^2 = (2 \times 3)^2 = 2^2 \times 3^2$

$6 = 2 \times 3$

Substitute these into the expression:

$\frac{3^{4} \;\times\; (2^2 \times 3)^{3} \;\times\; (2^2 \times 3^2)}{2^{5} \;\times\; (2 \times 3)^{3}}$

Use the power of a product and power of a power rules:

$\frac{3^{4} \;\times\; (2^2)^3 \times 3^3 \;\times\; 2^2 \times 3^2}{2^{5} \;\times\; 2^3 \times 3^3} = \frac{3^{4} \;\times\; 2^6 \times 3^3 \;\times\; 2^2 \times 3^2}{2^{5} \;\times\; 2^3 \times 3^3}$

Combine terms with the same base in the numerator and denominator using the product of powers rule:

Numerator: $2^6 \times 2^2 = 2^{6+2} = 2^8$

Numerator: $3^4 \times 3^3 \times 3^2 = 3^{4+3+2} = 3^9$

Denominator: $2^5 \times 2^3 = 2^{5+3} = 2^8$

Denominator: $3^3$

The expression becomes:

$\frac{2^8 \;\times\; 3^9}{2^8 \;\times\; 3^3}$

Use the quotient of powers rule:

$2^{8-8} \times 3^{9-3} = 2^0 \times 3^6$

Since $2^0 = 1$ (assuming base is not 0):

$= 1 \times 3^6 = 3^6$

Calculate the value of $3^6$:

$3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 9 \times 9 \times 9 = 81 \times 9 = 729$

The value is 729.

(e) Evaluate $\left( \frac{6 \;\times\; 10}{2^2 \;\times\; 5^3} \right)^{2} \times\frac{25}{27}$:

Simplify the expression inside the parenthesis first. Express composite bases in terms of prime factors:

$6 = 2 \times 3$

$10 = 2 \times 5$

Inside the parenthesis: $\frac{(2 \times 3) \times (2 \times 5)}{2^2 \times 5^3} = \frac{2^{1+1} \times 3 \times 5}{2^2 \times 5^3} = \frac{2^2 \times 3 \times 5^1}{2^2 \times 5^3}$

Simplify inside parenthesis using quotient of powers rule:

$= 2^{2-2} \times 3^1 \times 5^{1-3} = 2^0 \times 3^1 \times 5^{-2} = 1 \times 3 \times \frac{1}{5^2} = \frac{3}{5^2}$

Now apply the outer exponent (2):

$\left( \frac{3}{5^2} \right)^2 = \frac{3^2}{(5^2)^2} = \frac{3^2}{5^{2 \times 2}} = \frac{3^2}{5^4}$

Express the terms in the second fraction using prime factors:

$25 = 5^2$

$27 = 3^3$

Multiply the result from the parenthesis by the second fraction:

$\frac{3^2}{5^4} \times \frac{5^2}{3^3}$

Combine and use the quotient of powers rule:

$3^{2-3} \times 5^{2-4} = 3^{-1} \times 5^{-2}$

$= \frac{1}{3^1} \times \frac{1}{5^2} = \frac{1}{3} \times \frac{1}{25}$

$= \frac{1 \times 1}{3 \times 25} = \frac{1}{75}$

The value is $\frac{1}{75}$.

(f) Evaluate $\frac{15^{4} \;\times\; 18^{3}}{3^{3} \;\times\; 5^{2} \;\times\; 12^{2}}$:

Express the composite bases in terms of prime factors:

$15 = 3 \times 5$

$18 = 2 \times 9 = 2 \times 3^2$

$12 = 4 \times 3 = 2^2 \times 3$

Substitute these into the expression:

$\frac{(3 \times 5)^{4} \;\times\; (2 \times 3^2)^{3}}{3^{3} \;\times\; 5^{2} \;\times\; (2^2 \times 3)^{2}}$

Use the power of a product and power of a power rules:

$\frac{3^4 \times 5^4 \;\times\; 2^3 \times (3^2)^3}{3^3 \;\times\; 5^2 \;\times\; (2^2)^2 \times 3^2} = \frac{3^4 \times 5^4 \;\times\; 2^3 \times 3^6}{3^3 \;\times\; 5^2 \;\times\; 2^4 \times 3^2}$

Combine terms with the same base in the numerator and denominator using the product of powers rule:

Numerator: $2^3$, $3^4 \times 3^6 = 3^{4+6} = 3^{10}$, $5^4$

Denominator: $2^4$, $3^3 \times 3^2 = 3^{3+2} = 3^5$, $5^2$

The expression becomes:

$\frac{2^3 \;\times\; 3^{10} \;\times\; 5^4}{2^4 \;\times\; 3^{5} \;\times\; 5^2}$

Use the quotient of powers rule:

$2^{3-4} \times 3^{10-5} \times 5^{4-2}$

$= 2^{-1} \times 3^5 \times 5^2$

$= \frac{1}{2} \times 3^5 \times 5^2$

Calculate the values of the powers:

$3^5 = 243$

$5^2 = 25$

$= \frac{1}{2} \times 243 \times 25$

$= \frac{243 \times 25}{2} = \frac{6075}{2}$

The value is $\frac{6075}{2}$ or 3037.5.

(g) Evaluate $\frac{6^{4} \;\times\; 9^{2} \;\times\; 25^{3}}{3^{2} \;\times\; 4^{2} \;\times\; 15^{6}}$:

Express the composite bases in terms of prime factors:

$6 = 2 \times 3$

$9 = 3^2$

$25 = 5^2$

$4 = 2^2$

$15 = 3 \times 5$

Substitute these into the expression:

$\frac{(2 \times 3)^{4} \;\times\; (3^2)^{2} \;\times\; (5^2)^{3}}{3^{2} \;\times\; (2^2)^{2} \;\times\; (3 \times 5)^{6}}$

Use the power of a product and power of a power rules:

$\frac{2^4 \times 3^4 \;\times\; 3^{2 \times 2} \;\times\; 5^{2 \times 3}}{3^2 \;\times\; 2^{2 \times 2} \;\times\; 3^6 \times 5^6} = \frac{2^4 \times 3^4 \;\times\; 3^4 \;\times\; 5^6}{3^2 \;\times\; 2^4 \;\times\; 3^6 \times 5^6}$

Combine terms with the same base in the numerator and denominator using the product of powers rule:

Numerator: $2^4$, $3^4 \times 3^4 = 3^{4+4} = 3^8$, $5^6$

Denominator: $2^4$, $3^2 \times 3^6 = 3^{2+6} = 3^8$, $5^6$

The expression becomes:

$\frac{2^4 \;\times\; 3^8 \;\times\; 5^6}{2^4 \;\times\; 3^8 \;\times\; 5^6}$

Since the numerator and the denominator are identical (and non-zero), the value of the expression is 1.

Using the quotient of powers rule for each base:

$2^{4-4} \times 3^{8-8} \times 5^{6-6} = 2^0 \times 3^0 \times 5^0$

$= 1 \times 1 \times 1 = 1$

The value is 1.

First, we express the area of each desert in scientific notation (standard form).

• Kalahari, South Africa: $932,400 = 9.324 \times 10^5$ sq. km
• Thar, India: $932,400 = 9.324 \times 10^5$ sq. km
• Gibson, Australia: $155,400 = 1.554 \times 10^5$ sq. km
• Great Victoria, Australia: $647,500 = 6.475 \times 10^5$ sq. km
• Sahara, North Africa: $8,598,800 = 8.5988 \times 10^6$ sq. km

The areas in standard form are:

• Kalahari: $9.324 \times 10^5$
• Thar: $9.324 \times 10^5$
• Gibson: $1.554 \times 10^5$
• Great Victoria: $6.475 \times 10^5$
• Sahara: $8.5988 \times 10^6$

To arrange the areas in ascending order, we compare the exponents of 10 first, and then the decimal parts if the exponents are the same.

We have areas with exponent 5 and one area with exponent 6.

The area with exponent 6 ($8.5988 \times 10^6$) is the largest.

Now we compare the areas with exponent 5:

$1.554 \times 10^5$ (Gibson)

$6.475 \times 10^5$ (Great Victoria)

$9.324 \times 10^5$ (Kalahari)

$9.324 \times 10^5$ (Thar)

Arranging the decimal parts in ascending order: $1.554 < 6.475 < 9.324 = 9.324$.

So the order of these four is: Gibson, Great Victoria, Kalahari and Thar (Kalahari and Thar have the same area).

The Sahara desert with area $8.5988 \times 10^6 = 85.988 \times 10^5$ is the largest.

Arranging the deserts in ascending order of their size:

1. Gibson, Australia ($1.554 \times 10^5$ sq. km)
2. Great Victoria, Australia ($6.475 \times 10^5$ sq. km)
3. Kalahari, South Africa ($9.324 \times 10^5$ sq. km)
4. Thar, India ($9.324 \times 10^5$ sq. km)
5. Sahara, North Africa ($8.5988 \times 10^6$ sq. km)

(Note: Kalahari and Thar have the same area, so their order relative to each other does not matter in an ascending list of sizes).

The arrangement in ascending order of their size is:

Gibson, Great Victoria, Kalahari, Thar, Sahara

or

Gibson, Great Victoria, Thar, Kalahari, Sahara

First, we express the mass of each planet in scientific notation (standard form).

• Mercury: $330,000,000,000,000,000,000,000 = 3.3 \times 10^{23}$ kg
• Venus: $4,870,000,000,000,000,000,000,000 = 4.87 \times 10^{24}$ kg
• Earth: $5,980,000,000,000,000,000,000,000 = 5.98 \times 10^{24}$ kg
• Mars: $642,000,000,000,000,000,000,000 = 6.42 \times 10^{23}$ kg
• Jupiter: $1,900,000,000,000,000,000,000,000,000 = 1.9 \times 10^{27}$ kg
• Saturn: $569,000,000,000,000,000,000,000,000 = 5.69 \times 10^{26}$ kg
• Uranus: $86,900,000,000,000,000,000,000,000 = 8.69 \times 10^{25}$ kg
• Neptune: $102,000,000,000,000,000,000,000,000 = 1.02 \times 10^{26}$ kg
• Pluto: $13,100,000,000,000,000,000,000 = 1.31 \times 10^{22}$ kg

The masses in standard form are:

• Mercury: $3.3 \times 10^{23}$ kg
• Venus: $4.87 \times 10^{24}$ kg
• Earth: $5.98 \times 10^{24}$ kg
• Mars: $6.42 \times 10^{23}$ kg
• Jupiter: $1.9 \times 10^{27}$ kg
• Saturn: $5.69 \times 10^{26}$ kg
• Uranus: $8.69 \times 10^{25}$ kg
• Neptune: $1.02 \times 10^{26}$ kg
• Pluto: $1.31 \times 10^{22}$ kg

To arrange the planets in descending order of their size (mass), we compare the exponents of 10. The larger the exponent, the larger the number. If the exponents are the same, we compare the decimal parts.

Comparing the exponents: $23, 24, 24, 23, 27, 26, 25, 26, 22$.

The exponents in descending order are: $27, 26, 26, 25, 24, 24, 23, 23, 22$.

Now, we arrange the planets based on their masses in descending order:

1. Jupiter: $1.9 \times 10^{27}$ kg
2. Saturn: $5.69 \times 10^{26}$ kg
3. Neptune: $1.02 \times 10^{26}$ kg
4. Uranus: $8.69 \times 10^{25}$ kg
5. Earth: $5.98 \times 10^{24}$ kg
6. Venus: $4.87 \times 10^{24}$ kg
7. Mars: $6.42 \times 10^{23}$ kg
8. Mercury: $3.3 \times 10^{23}$ kg
9. Pluto: $1.31 \times 10^{22}$ kg

The planets arranged in descending order of their size (mass) are:

Jupiter, Saturn, Neptune, Uranus, Earth, Venus, Mars, Mercury, Pluto.

We need to express the given values in seconds in scientific notation ($a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer).

1. 1 minute = 60 seconds

$60 = 6.0 \times 10^1$ seconds

2. 1 Hour = 3,600 seconds

$3,600 = 3.6 \times 10^3$ seconds

3. 1 Day = 3,600 seconds

$3,600 = 3.6 \times 10^3$ seconds

4. 1 Month = 86,400 seconds

$86,400 = 8.64 \times 10^4$ seconds

5. 1 Year = 32,000,000 seconds

$32,000,000 = 3.2 \times 10^7$ seconds

6. 10 Years = 3,20,000,000 seconds (which is 320,000,000)

$320,000,000 = 3.2 \times 10^8$ seconds

The values in scientific notation are:

• 1 minute: $6.0 \times 10^1$
• 1 Hour: $3.6 \times 10^3$
• 1 Day: $3.6 \times 10^3$
• 1 Month: $8.64 \times 10^4$
• 1 Year: $3.2 \times 10^7$
• 10 Years: $3.2 \times 10^8$

Arranging these values in ascending order of their size, we compare the exponents of 10 first, and then the decimal parts if the exponents are the same.

The exponents are: $1, 3, 3, 4, 7, 8$. These are already in ascending order.

Comparing the values with exponent 3: $3.6 \times 10^3$ (1 Hour) and $3.6 \times 10^3$ (1 Day). They have the same value.

Comparing the values with exponent 7 and 8: $3.2 \times 10^7 < 3.2 \times 10^8$.

Thus, the values in ascending order are:

$6.0 \times 10^1$, $3.6 \times 10^3$, $3.6 \times 10^3$, $8.64 \times 10^4$, $3.2 \times 10^7$, $3.2 \times 10^8$.

Replacing with the units, the arrangement in ascending order of size is:

1 minute, 1 Hour, 1 Day, 1 Month, 1 Year, 10 Years.

(Note: Based on the values provided in the table, 1 Hour and 1 Day have the same value, so their relative order does not matter in an ascending list of sizes).

Given:

Area covered with water = 361,419,000 sq km

Area covered by land = 148,647,000 sq km

To Find:

Approximate ratio of water area to land area, using scientific notation.

Solution:

First, we express the given areas in scientific notation ($a \times 10^n$, where $1 \leq |a| < 10$).

Area covered with water = 361,419,000 sq km

Moving the decimal point 8 places to the left:

$361,419,000 = 3.61419 \times 10^8$ sq km

Area covered by land = 148,647,000 sq km

Moving the decimal point 8 places to the left:

$148,647,000 = 1.48647 \times 10^8$ sq km

The ratio of area covered with water to area covered by land is:

Ratio = $\frac{\text{Area covered with water}}{\text{Area covered by land}}$

Ratio = $\frac{3.61419 \times 10^8}{1.48647 \times 10^8}$

Using the property of exponents $\frac{10^m}{10^n} = 10^{m-n}$, we simplify the power of 10 terms:

$\frac{10^8}{10^8} = 10^{8-8} = 10^0 = 1$

So the ratio simplifies to:

Ratio = $\frac{3.61419}{1.48647}$

To find the approximate ratio, we perform the division of the decimal parts. We can approximate these numbers to a few decimal places for simplicity before dividing, or perform the division and then approximate the result.

Let's approximate $3.61419$ to $3.61$ and $1.48647$ to $1.49$.

Approximate Ratio $\approx \frac{3.61}{1.49}$

Performing the division:

$\frac{3.61}{1.49} \approx 2.4228$

Rounding to one decimal place, the approximate ratio is 2.4.

Alternatively, using slightly simpler approximations from the scientific notation: $3.6 \times 10^8$ and $1.5 \times 10^8$.

Approximate Ratio $\approx \frac{3.6 \times 10^8}{1.5 \times 10^8} = \frac{3.6}{1.5} = \frac{36}{15} = \frac{12}{5} = 2.4$

The approximate ratio of area covered with water to area covered by land is 2.4.

Given the equation:

$2^{n + 2} – 2^{n + 1} + 2^{n} = c \times 2^{n}$

We need to simplify the left side (LHS) of the equation. We can express each term in the LHS as a product involving $2^n$ using the property $a^{m+n} = a^m \times a^n$:

$2^{n+2} = 2^n \times 2^2$

$2^{n+1} = 2^n \times 2^1$

$2^n = 2^n \times 1$

Substitute these expressions back into the LHS:

LHS = $(2^n \times 2^2) - (2^n \times 2^1) + (2^n \times 1)$

Now, we can factor out the common term $2^n$ from each term in the LHS:

LHS = $2^n (2^2 - 2^1 + 1)$

Evaluate the expression inside the parenthesis:

$2^2 - 2^1 + 1 = 4 - 2 + 1 = 2 + 1 = 3$

So, the simplified LHS is:

LHS = $2^n \times 3$

Now, equate the simplified LHS to the RHS of the original equation:

$2^n \times 3 = c \times 2^n$

To find the value of $c$, we can divide both sides of the equation by $2^n$ (assuming $2^n \neq 0$, which is always true for any integer $n$):

$\frac{2^n \times 3}{2^n} = \frac{c \times 2^n}{2^n}$

$3 = c$

Thus, the value of c is 3.

Given:

1 light year = 9,460,000,000,000 km

Average distance between Earth and Sun = $1.496 \times 10^8$ km

To Find:

(a) One light year in scientific notation.

(b) Comparison between Earth-Sun distance and one light year.

Solution:

(a) Expressing one light year in scientific notation:

The given distance is 9,460,000,000,000 km.

To write this in scientific notation, we need to express it in the form $a \times 10^n$, where $1 \leq a < 10$.

We move the decimal point from the end of the number to the left until it is after the first non-zero digit (9).

The number of places the decimal point is moved is 12.

$9,460,000,000,000 = 9.46 \times 10^{12}$

So, 1 light year = $9.46 \times 10^{12}$ km.

(b) Comparing the distance between Earth and the Sun with one light year:

The average distance between Earth and Sun is $1.496 \times 10^8$ km.

One light year is $9.46 \times 10^{12}$ km.

To compare these two numbers, we look at the exponents of 10.

The exponent for the Earth-Sun distance is 8.

The exponent for one light year is 12.

Since $8 < 12$, the number with the exponent 8 is smaller than the number with the exponent 12.

$1.496 \times 10^8 < 9.46 \times 10^{12}$

Therefore, the distance between Earth and the Sun is less than one light year.

(a) One light year in scientific notation is $9.46 \times 10^{12}$ km.

(b) The distance between Earth and the Sun is less than one light year.

Given:

The formula for the number of diagonals of an n-sided figure is $\frac{1}{2}(n^2 - 3n)$.

The figure is a 6-sided figure (hexagon).

To Find:

The number of diagonals for a 6-sided figure.

Solution:

For a 6-sided figure, the number of sides $n = 6$.

Substitute $n=6$ into the given formula:

Number of diagonals = $\frac{1}{2}(n^2 - 3n)$

= $\frac{1}{2}((6)^2 - 3 \times 6)$

First, calculate the terms inside the parenthesis:

$6^2 = 6 \times 6 = 36$

$3 \times 6 = 18$

Substitute these values back into the formula:

= $\frac{1}{2}(36 - 18)$

= $\frac{1}{2}(18)$

Now, multiply by $\frac{1}{2}$:

= $\frac{18}{2} = 9$

Thus, the number of diagonals for a 6-sided figure (hexagon) is 9.

Given:

Initial number of bacteria = 1

Doubling time = 20 minutes

Total time = 6 hours

To Find:

The number of bacteria after 6 hours, expressed in exponential form and evaluated.

Solution:

The number of bacteria doubles every 20 minutes. This means that after $k$ doubling periods, the initial number of bacteria is multiplied by $2^k$.

First, we need to find the number of 20-minute intervals in 6 hours. We convert the total time to minutes:

1 hour = 60 minutes

6 hours = $6 \times 60$ minutes = 360 minutes

Next, we find the number of 20-minute intervals in 360 minutes:

Number of intervals = $\frac{\text{Total time}}{\text{Doubling time}}$

= $\frac{360 \text{ minutes}}{20 \text{ minutes/interval}} = \frac{360}{20} = 18$ intervals

The number of bacteria after 18 doubling periods, starting with 1 bacterium, will be $1 \times 2^{18}$.

The answer in exponential form is $2^{18}$.

Now, we evaluate $2^{18}$:

$2^{18} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$2^{18} = 2^{10} \times 2^8$

We know that $2^{10} = 1024$ and $2^8 = 256$.

$2^{18} = 1024 \times 256$

$\begin{array}{cc}& & 1 & 0 & 2 & 4 \\ \times & & & & 2 & 5 & 6 \\ \hline && 6 & 1 & 4 & 4 \\ & 5 & 1 & 2 & 0 & \times \\ 2 & 0 & 4 & 8 & \times & \times \\ \hline 2 & 6 & 2 & 1 & 4 & 4 \\ \hline \end{array}$

The value of $2^{18}$ is 262,144.

There will be $2^{18}$ or 262,144 bacteria in 6 hours.

Given:

The amount of blubber weight is given by the expression $2^2 \times 3^2 \times 5 \times 17$ kg.

To Evaluate:

The value of the given expression.

Solution:

We need to calculate the value of the expression $2^2 \times 3^2 \times 5 \times 17$.

First, evaluate the terms with exponents:

$2^2 = 2 \times 2 = 4$

$3^2 = 3 \times 3 = 9$

Substitute these values back into the expression:

Value = $4 \times 9 \times 5 \times 17$

Now, multiply the numbers:

$4 \times 9 = 36$

$36 \times 5 = 180$

$180 \times 17$

Perform the multiplication of 180 by 17:

$\begin{array}{cc}& & 1 & 8 & 0 \\ \times & & & 1 & 7 \\ \hline && 1 & 2 & 6 & 0 \\ & 1 & 8 & 0 & \times \\ \hline 3 & 0 & 6 & 0 \\ \hline \end{array}$

The value of the expression is 3060.

Since the amount is given in kg, the evaluated amount is 3060 kg.

The evaluated amount of the blubber weight is 3060 kg.

Given:

Diameter of a typical red blood cell = $7 \times 10^{-6}$ metres

Diameter of a typical platelet = $2.33 \times 10^{-6}$ metres

To Determine:

Which has a greater diameter, a red blood cell or a platelet.

Solution:

We are given the diameters of the red blood cell and the platelet in scientific notation:

Red blood cell diameter = $7 \times 10^{-6}$ m

Platelet diameter = $2.33 \times 10^{-6}$ m

To compare numbers written in scientific notation ($a \times 10^n$), we first compare the exponents of 10. If the exponents are the same, we then compare the values of $a$ (the number between 1 and 10).

In this case, both numbers have the same exponent of 10, which is $-6$.

Now, we compare the decimal parts (the coefficients $a$):

For the red blood cell, $a_1 = 7$.

For the platelet, $a_2 = 2.33$.

Comparing the values of $a_1$ and $a_2$:

$7$ compared to $2.33$

Since $7 > 2.33$, the number with the coefficient 7 is greater.

Therefore, $7 \times 10^{-6}$ metres is greater than $2.33 \times 10^{-6}$ metres.

A red blood cell has a greater diameter than a platelet.

Given:

A googol is the number 1 followed by 100 zeroes.

To Find:

(a) A googol written as a power.

(b) A googol times a googol written as a power.

Solution:

(a) A number consisting of the digit 1 followed by $n$ zeroes is equal to $10^n$.

A googol is the number 1 followed by 100 zeroes.

Therefore, a googol can be written as a power of 10 with an exponent equal to the number of zeroes.

Googol = $10^{100}$

A googol written as a power is $10^{100}$.

(b) We need to calculate a googol times a googol and write the result as a power.

Googol $\times$ Googol = $10^{100} \times 10^{100}$

Using the property of exponents $a^m \times a^n = a^{m+n}$, where $a=10$, $m=100$, and $n=100$:

$10^{100} \times 10^{100} = 10^{100+100}$

$= 10^{200}$

A googol times a googol written as a power is $10^{200}$.

The student is evaluating the expression $\frac{3^5}{9^5}$.

The student claims that $\frac{3^5}{9^5} = \frac{1}{3}$.

Let's correctly evaluate the expression $\frac{3^5}{9^5}$.

The expression has the form $\frac{a^m}{b^m}$, where the exponents are the same but the bases are different. We can use the property $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.

$\frac{3^5}{9^5} = \left( \frac{3}{9} \right)^5$

Now, simplify the fraction inside the parenthesis:

$\frac{3}{9} = \frac{1}{3}$

Substitute the simplified fraction back into the expression:

$\left( \frac{1}{3} \right)^5$

Now, evaluate the power:

$\left( \frac{1}{3} \right)^5 = \frac{1^5}{3^5} = \frac{1}{3 \times 3 \times 3 \times 3 \times 3} = \frac{1}{243}$

The correct value of $\frac{3^5}{9^5}$ is $\frac{1}{243}$, not $\frac{1}{3}$.

The student's mistake is that they simplified the fraction inside the parenthesis, $\frac{3}{9}$, to $\frac{1}{3}$, but they failed to apply the exponent 5 to this simplified fraction. They effectively calculated $(\frac{3}{9})^1$ instead of $(\frac{3}{9})^5$. The exponent 5 must be applied to the entire simplified base $\left(\frac{1}{3}\right)$.