Classwise Concept with Examples
6th	7th	8th	9th	10th	11th	12th

Class 7th Chapters
1. Integers	2. Fractions and Decimals	3. Data Handling
4. Simple Equations	5. Lines and Angles	6. The Triangle and its Properties
7. Congruence of Triangles	8. Comparing Quantities	9. Rational Numbers
10. Practical Geometry	11. Perimeter and Area	12. Algebraic Expressions
13. Exponents and Powers	14. Symmetry	15. Visualising Solid Shapes

Content On This Page
Exponents	Laws of Exponents	Uses of Exponents

Chapter 13 Exponents and Powers (Concepts)

Welcome to this chapter exploring the powerful concept of Exponents and Powers. As we delve deeper into mathematics, particularly when dealing with extremely large or incredibly small numbers frequently encountered in science and engineering, we often find that repeated multiplication becomes cumbersome. This chapter introduces a highly efficient and concise mathematical notation to represent such repeated multiplications, streamlining calculations and enhancing our ability to express numerical magnitudes effectively. This notation revolves around the idea of an exponent, also commonly referred to as a power or index.

We define the exponential notation as $a^n$. In this notation, 'a' is called the base, which is the number being multiplied, and 'n' is the exponent (or power), indicating how many times the base 'a' is multiplied by itself. The expression $a^n$ is typically read as "a raised to the power of n" or simply "a to the power n". For instance, instead of writing $5 \times 5 \times 5 \times 5$, we can compactly write this as $5^4$, where $5$ is the base and $4$ is the exponent. Students will practice converting numbers expressed as repeated multiplications into this exponential form and, conversely, evaluating exponential expressions to find their numerical value (e.g., calculating $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$).

The true power of this notation unfolds when we learn the Laws of Exponents. These fundamental rules provide systematic ways to simplify calculations involving numbers expressed in exponential form, especially when dealing with multiplication, division, or nested powers. Mastering these laws is crucial:

Product Rule: When multiplying exponential terms with the same base, we add their exponents: $a^m \times a^n = a^{m+n}$. (Example: $3^2 \times 3^4 = 3^{2+4} = 3^6$)
Quotient Rule: When dividing exponential terms with the same base, we subtract the exponent of the divisor from the exponent of the dividend: $\frac{a^m}{a^n} = a^{m-n}$, provided $a \neq 0$. (Example: $\frac{7^5}{7^2} = 7^{5-2} = 7^3$)
Power of a Power Rule: When raising an exponential term to another power, we multiply the exponents: $(a^m)^n = a^{m \times n}$. (Example: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$)
Product to a Power Rule: When a product of different bases is raised to a power, we can distribute the exponent to each base: $(a \times b)^m = a^m \times b^m$. (Example: $(2 \times 5)^3 = 2^3 \times 5^3$)
Quotient to a Power Rule: When a quotient of different bases is raised to a power, we distribute the exponent to both the numerator and the denominator: $(\frac{a}{b})^m = \frac{a^m}{b^m}$, provided $b \neq 0$. (Example: $(\frac{4}{9})^2 = \frac{4^2}{9^2}$)

Furthermore, two special exponent definitions are essential:

Zero Exponent: Any non-zero number raised to the power of zero is defined as equal to 1: $a^0 = 1$, for any $a \neq 0$.
Negative Exponent: A base raised to a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent: $a^{-n} = \frac{1}{a^n}$, for any $a \neq 0$. (Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$)

These laws and definitions are instrumental in simplifying complex algebraic and numerical expressions involving powers. Another significant application introduced is expressing very large or very small numbers using Standard Form, also widely known as Scientific Notation. This method represents a number as the product of two parts: a decimal number $k$ such that $1 \le k < 10$, and an integer power of 10 ($10^n$). The format is $k \times 10^n$. For example, the distance to the sun (approx. 150,000,000 km) can be written as $1.5 \times 10^8$ km, and the diameter of a hydrogen atom (approx. 0.0000000001 m) can be written as $1 \times 10^{-10}$ m. This notation drastically simplifies writing, comparing, and calculating with such extreme values, proving indispensable in scientific fields.

Exponents

Sometimes, we encounter situations where a number needs to be multiplied by itself multiple times. For instance, consider multiplying the number 2 by itself 6 times: $2 \times 2 \times 2 \times 2 \times 2 \times 2$. Writing this repeated multiplication every time can be lengthy and difficult to manage, especially if the number is multiplied by itself many more times. To simplify and shorten the representation of such repeated multiplication, we use a mathematical notation called exponents or powers.

What is an Exponent?

An exponent, also known as a power or index, is a mathematical notation that indicates how many times a base number is multiplied by itself.

An expression written using exponents is called an exponential form. It generally looks like $a^n$.

In the expression $a^n$:

The number '$a$' is called the base. It is the number that is being multiplied repeatedly.
The number '$n$' is called the exponent or power. It tells us how many times the base is multiplied by itself.

The expression $a^n$ is read as "a raised to the power n", "a to the nth power", or simply "a to the n".

For example, in the expression $2^6$, the base is 2 and the exponent is 6. This means the number 2 is multiplied by itself 6 times:

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

$ = 4 \times 2 \times 2 \times 2 \times 2 $

$ = 8 \times 2 \times 2 \times 2 $

$ = 16 \times 2 \times 2 $

$ = 32 \times 2 $

$ = 64 $

The value of $2^6$ is 64. 64 is the result of raising the base 2 to the exponent 6.

Understanding Powers

Let's look at a few more examples to understand how exponents work:

$10^2 = 10 \times 10 = 100$. This is read as "10 raised to the power 2" or "10 squared".
$5^3 = 5 \times 5 \times 5 = 125$. This is read as "5 raised to the power 3" or "5 cubed".
$a^4 = a \times a \times a \times a$. This is read as "a raised to the power 4" or "a to the fourth power".
$(-3)^2 = (-3) \times (-3)$. When we multiply two negative numbers, the result is positive. So, $(-3)^2 = 9$.
$(-3)^3 = (-3) \times (-3) \times (-3)$. We first multiply $(-3) \times (-3) = 9$. Then multiply 9 by the remaining $(-3)$. So, $9 \times (-3) = -27$. Hence, $(-3)^3 = -27$.

From the examples with a negative base, we can observe a pattern:

When a negative base is raised to an even exponent (like 2, 4, 6, ...), the result is positive. This is because the negative signs cancel out in pairs.
When a negative base is raised to an odd exponent (like 1, 3, 5, ...), the result is negative. This is because after cancelling out pairs of negative signs, one negative sign remains.

Example 1. Express the following in exponential form:

(a) $7 \times 7 \times 7 \times 7$

(b) $p \times p \times p \times p \times p$

Answer:

(a) In the expression $7 \times 7 \times 7 \times 7$, the base is 7, and it is multiplied by itself 4 times. So, the exponent is 4. The exponential form is $7^4$.

(b) In the expression $p \times p \times p \times p \times p$, the base is $p$, and it is multiplied by itself 5 times. So, the exponent is 5. The exponential form is $p^5$.

(c) In the expression $3 \times 3 \times 5 \times 5 \times 5$, we have two different bases being multiplied. The number 3 is multiplied by itself 2 times, which can be written as $3^2$. The number 5 is multiplied by itself 3 times, which can be written as $5^3$. Therefore, the combined exponential form is the product of these two exponential forms: $3^2 \times 5^3$.

Example 2. Find the value of:

(a) $3^4$

(b) $(-2)^5$

Answer:

(a) $3^4$ means 3 multiplied by itself 4 times.

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

$ = (3 \times 3) \times (3 \times 3) $

$ = 9 \times 9 $

$ = 81 $

(b) $(-2)^5$ means -2 multiplied by itself 5 times.

$(-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) $

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

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

$ = 16 \times (-2) $

$ = -32 $

Alternatively, we know that a negative base raised to an odd power is negative. So, $(-2)^5 = -(2^5) = -(2 \times 2 \times 2 \times 2 \times 2) = -(32) = -32$.

$(1.5)^2 = 1.5 \times 1.5$

$ = 2.25 $

Example 3. Identify the base and exponent in each of the following expressions and write their expanded form:

(a) $10^5$

(b) $(-4)^3$

Answer:

(a) In $10^5$, the base is 10 and the exponent is 5.

Expanded form: $10 \times 10 \times 10 \times 10 \times 10$

(b) In $(-4)^3$, the base is -4 and the exponent is 3.

Expanded form: $(-4) \times (-4) \times (-4)$

Expanded form: $\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}$

Example 4. Express the following numbers in exponential form:

(a) 81 (using base 3)

(b) 128 (using base 2)

Answer:

To express a number in exponential form with a given base, we repeatedly divide the number by the base until we get 1. The number of times we divide is the exponent.

(a) Expressing 81 using base 3:

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

We divided 81 by 3, 4 times. So, $81 = 3 \times 3 \times 3 \times 3$.

Exponential form: $3^4$

(b) Expressing 128 using base 2:

$\begin{array}{c|cc} 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \end{array}$

We divided 128 by 2, 7 times. So, $128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$.

Exponential form: $2^7$

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

We divided 243 by 3, 5 times. So, $243 = 3 \times 3 \times 3 \times 3 \times 3$.

Exponential form: $3^5$

Example 5. Compare the following numbers:

(a) $2^3$ and $3^2$

(b) $5^4$ and $4^5$

Answer:

To compare exponential numbers, we need to find their actual values.

(a) Compare $2^3$ and $3^2$:

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

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

Since $8 < 9$, we have $2^3 < 3^2$.

(b) Compare $5^4$ and $4^5$:

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

$4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 16 \times 4 \times 4 \times 4 = 64 \times 4 \times 4 = 256 \times 4 = 1024$

Since $625 < 1024$, we have $5^4 < 4^5$.

Example 6. Find the value of:

(a) $(-5)^3$

(b) $(-1)^6$

Answer:

(a) $(-5)^3 = (-5) \times (-5) \times (-5)$

$ = ((-5) \times (-5)) \times (-5) $

$ = (25) \times (-5) $

$ = -125 $

Alternatively, negative base to an odd power is negative: $(-5)^3 = -(5^3) = -(5 \times 5 \times 5) = -125$.

(b) $(-1)^6 = (-1) \times (-1) \times (-1) \times (-1) \times (-1) \times (-1)$

Since the exponent (6) is even, the result will be positive.

$ = 1 \times 1 \times 1 = 1 $

In general, $(-1)^{\text{even number}} = 1$ and $(-1)^{\text{odd number}} = -1$.

$ = (0.1 \times 0.1) \times (0.1 \times 0.1) $

$ = (0.01) \times (0.01) $

$ = 0.0001 $

Alternatively, $(0.1)^4 = \left(\frac{1}{10}\right)^4 = \frac{1^4}{10^4} = \frac{1}{10000} = 0.0001$.

Laws of Exponents

Now that we understand what exponents are and how to represent repeated multiplication in exponential form, let's learn about some important rules that help us work with exponents more efficiently. These rules are called the Laws of Exponents. They are applicable when dealing with powers that have the same base or the same exponent.

Law 1: Multiplying Powers with the Same Base

When we multiply exponential expressions that have the same base, we keep the base the same and add the exponents.

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

... (i)

Here, '$a$' is any non-zero integer (or rational number), and '$m$' and '$n$' are whole numbers.

Derivation/Explanation:

Let's consider an example, say $2^3 \times 2^4$.

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

(Definition of exponent)

$2^4 = 2 \times 2 \times 2 \times 2$

(Definition of exponent)

So, $2^3 \times 2^4$ means multiplying the expanded forms:

$2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)$

By associative property of multiplication, we can remove the brackets:

$ = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $

Now, we have the base 2 multiplied by itself a total of $3 + 4 = 7$ times. By the definition of exponents, this can be written as $2^7$.

$ = 2^7 $

Thus, $2^3 \times 2^4 = 2^{3+4} = 2^7$. This example illustrates the general rule $a^m \times a^n = a^{m+n}$.

Law 2: Dividing Powers with the Same Base

When we divide exponential expressions that have the same base, we keep the base the same and subtract the exponent of the denominator from the exponent of the numerator.

Rule: $a^m \div a^n = a^{m-n}$

... (ii)

Here, '$a$' is any non-zero integer (or rational number), and '$m$' and '$n$' are whole numbers with $m > n$. (We will discuss the case where $m \le n$ later).

Derivation/Explanation:

Let's consider an example, say $5^6 \div 5^2$. This can be written as $\frac{5^6}{5^2}$.

Using the definition of exponents, we can expand the numerator and the denominator:

$\frac{5^6}{5^2} = \frac{5 \times 5 \times 5 \times 5 \times 5 \times 5}{5 \times 5}$

Now, we can cancel out the common factors in the numerator and the denominator. For every 5 in the denominator, we cancel one 5 in the numerator:

$ = \frac{\cancel{5} \times \cancel{5} \times 5 \times 5 \times 5 \times 5}{\cancel{5} \times \cancel{5}} $

After cancellation, we are left with 5 multiplied by itself $6 - 2 = 4$ times in the numerator:

$ = 5 \times 5 \times 5 \times 5 $

Using the definition of exponents, this is $5^4$.

$ = 5^4 $

Thus, $5^6 \div 5^2 = 5^{6-2} = 5^4$. This example illustrates the general rule $a^m \div a^n = a^{m-n}$ when $m > n$.

The case where $m=n$ is covered in Law 6 (Zero Exponent).

Law 3: Power of a Power

When an exponential expression (a power) is raised to another power, we keep the base the same and multiply the exponents.

Rule: $(a^m)^n = a^{m \times n}$

... (iii)

Here, '$a$' is any non-zero integer (or rational number), and '$m$' and '$n$' are whole numbers.

Derivation/Explanation:

Let's consider an example, say $(3^2)^3$.

The base here is $3^2$, and it is raised to the power of 3. This means the base $3^2$ is multiplied by itself 3 times:

$(3^2)^3 = (3^2) \times (3^2) \times (3^2) $

Now, let's expand $3^2$ which is $3 \times 3$:

$ = (3 \times 3) \times (3 \times 3) \times (3 \times 3) $

Removing the brackets, we have 3 multiplied by itself a total of $2+2+2 = 3 \times 2 = 6$ times:

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

Using the definition of exponents, this is $3^6$.

$ = 3^6 $

Thus, $(3^2)^3 = 3^{2 \times 3} = 3^6$. This example illustrates the general rule $(a^m)^n = a^{m \times n}$.

Note that $(a^m)^n = a^{m \times n}$ is generally equal to $(a^n)^m = a^{n \times m}$ because multiplication of whole numbers is commutative ($m \times n = n \times m$).

Law 4: Multiplying Powers with the Same Exponent but Different Bases

When we multiply exponential expressions that have different bases but the same exponent, we multiply the bases and keep the exponent the same.

Rule: $a^m \times b^m = (ab)^m$

... (iv)

Here, '$a$' and '$b$' are any non-zero integers (or rational numbers), and '$m$' is a whole number.

Derivation/Explanation:

Let's consider an example, say $2^3 \times 5^3$.

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

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

So, $2^3 \times 5^3$ means multiplying the expanded forms:

$2^3 \times 5^3 = (2 \times 2 \times 2) \times (5 \times 5 \times 5) $

Using the commutative and associative properties of multiplication, we can rearrange and group the terms:

$ = (2 \times 5) \times (2 \times 5) \times (2 \times 5) $

Inside each bracket, $2 \times 5 = 10$. So, we have 10 multiplied by itself 3 times:

$ = 10 \times 10 \times 10 $

Using the definition of exponents, this is $10^3$.

$ = 10^3 $

Thus, $2^3 \times 5^3 = (2 \times 5)^3 = 10^3$. This example illustrates the general rule $a^m \times b^m = (ab)^m$.

Law 5: Dividing Powers with the Same Exponent but Different Bases

When we divide exponential expressions that have different bases but the same exponent, we divide the bases and keep the exponent the same.

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

... (v)

Here, '$a$' and '$b$' are any non-zero integers (or rational numbers), and '$m$' is a whole number. Note that $b$ cannot be zero since it is in the denominator.

Derivation/Explanation:

Let's consider an example, say $6^4 \div 3^4$. This can be written as $\frac{6^4}{3^4}$.

Using the definition of exponents, we can expand the numerator and the denominator:

$\frac{6^4}{3^4} = \frac{6 \times 6 \times 6 \times 6}{3 \times 3 \times 3 \times 3}$

We can rewrite this fraction as a product of fractions:

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

Now, simplify each fraction $\frac{6}{3} = 2$:

$ = 2 \times 2 \times 2 \times 2 $

Using the definition of exponents, this is $2^4$.

$ = 2^4 $

We can see that $2 = 6 \div 3$ or $\frac{6}{3}$. So, $6^4 \div 3^4 = \left(\frac{6}{3}\right)^4 = 2^4$. This example illustrates the general rule $a^m \div b^m = \left(\frac{a}{b}\right)^m$.

Law 6: Zero Exponent

Any non-zero number (base) raised to the power of zero is always equal to 1.

Rule: $a^0 = 1$

... (vi)

Here, '$a$' is any non-zero number. Note that $0^0$ is undefined.

Derivation/Explanation:

We can understand this law using Law 2 (Dividing Powers with the Same Base). Let's consider $a^m \div a^m$.

Using Law 2, where we subtract the exponents ($m-n$), when $m=n$, we get:

$a^m \div a^m = a^{m-m} = a^0$

However, we also know that any non-zero number divided by itself is equal to 1. So, for $a \neq 0$:

$\frac{a^m}{a^m} = 1$

Since both expressions represent the same division, we can conclude that:

$a^0 = 1$

(For any $a \neq 0$)

Examples:

$7^0 = 1$
$(-5)^0 = 1$
$(100)^0 = 1$
$(x+y)^0 = 1$ (provided $x+y \neq 0$)

The expression $0^0$ is special and is generally considered undefined in mathematics at this level.

Law 7: Exponent One

Any number (base) raised to the power of 1 is equal to the base itself.

Rule: $a^1 = a$

... (vii)

Here, '$a$' is any number (integer or rational number).

Explanation:

This law directly follows from the definition of an exponent. The exponent tells us how many times the base is multiplied by itself. If the exponent is 1, it means the base is multiplied by itself only one time, which simply results in the base itself.

For example, $8^1$ means 8 is multiplied by itself 1 time, which is just 8. So, $8^1 = 8$.

Examples:

$(-10)^1 = -10$
$x^1 = x$
$50^1 = 50$

This law is often implicitly used when a number is written without an exponent, as it's understood that the exponent is 1.

Example 1. Simplify and write in exponential form:

(a) $2^5 \times 2^3$

(b) $7^8 \div 7^2$

(d) $4^3 \times 5^3$

(e) $\frac{10^5}{5^5}$

(f) $10^0$

(g) $p^5 \times p^8$

(h) $y^{10} \div y^7$

(i) $((a^3)^2)^4$

Answer:

(a) $2^5 \times 2^3$

$2^5 \times 2^3 = 2^{5+3}$

[Using Law 1: $a^m \times a^n = a^{m+n}$]

$ = 2^8 $

(b) $7^8 \div 7^2$

$7^8 \div 7^2 = 7^{8-2}$

[Using Law 2: $a^m \div a^n = a^{m-n}$]

$ = 7^6 $

$(6^2)^4 = 6^{2 \times 4}$

[Using Law 3: $(a^m)^n = a^{m \times n}$]

$ = 6^8 $

(d) $4^3 \times 5^3$

$4^3 \times 5^3 = (4 \times 5)^3$

[Using Law 4: $a^m \times b^m = (ab)^m$]

$ = 20^3 $

(e) $\frac{10^5}{5^5}$

$\frac{10^5}{5^5} = \left(\frac{10}{5}\right)^5$

[Using Law 5: $a^m \div b^m = (\frac{a}{b})^m$]

$ = 2^5 $

(f) $10^0$

$10^0 = 1$

[Using Law 6: $a^0 = 1$ (for $a \neq 0$)]

(g) $p^5 \times p^8$

$p^5 \times p^8 = p^{5+8}$

[Using Law 1: $a^m \times a^n = a^{m+n}$]

$ = p^{13} $

(h) $y^{10} \div y^7$

$y^{10} \div y^7 = y^{10-7}$

[Using Law 2: $a^m \div a^n = a^{m-n}$]

$ = y^3 $

(i) $((a^3)^2)^4$

$((a^3)^2)^4 = (a^{3 \times 2})^4$

[Using Law 3: $(a^m)^n = a^{m \times n}$]

$ = (a^6)^4$

$ = a^{6 \times 4} $

[Using Law 3 again]

$ = a^{24} $

Example 2. Simplify:

(a) $(3^2 \times 3^5) \div 3^4$

(b) $m^4 \times m^5 \times m^3$

Answer:

(a) $(3^2 \times 3^5) \div 3^4$

$ (3^2 \times 3^5) \div 3^4 = 3^{2+5} \div 3^4 $

[Using Law 1 for the multiplication in the bracket]

$ = 3^7 \div 3^4 $

$ = 3^{7-4} $

[Using Law 2 for the division]

$ = 3^3 $

We can also find the value: $3^3 = 3 \times 3 \times 3 = 27$.

(b) $m^4 \times m^5 \times m^3$

$ m^4 \times m^5 \times m^3 = m^{4+5} \times m^3 $

[Using Law 1 for the first two terms]

$ = m^9 \times m^3 $

$ = m^{9+3} $

[Using Law 1 again]

$ = m^{12} $

Note: We can extend Law 1 to any number of terms with the same base: $a^m \times a^n \times a^p = a^{m+n+p}$. So, $m^4 \times m^5 \times m^3 = m^{4+5+3} = m^{12}$.

We can rearrange the terms using the commutative property of multiplication to group terms with the same base:

$ 5^2 \times 3^2 \times 5^3 \times 3^4 = (5^2 \times 5^3) \times (3^2 \times 3^4) $

Now, apply Law 1 to each group:

$ = 5^{2+3} \times 3^{2+4} $

[Using Law 1]

$ = 5^5 \times 3^6 $

This expression cannot be simplified further using these laws because the bases (5 and 3) are different, and the exponents (5 and 6) are also different.

Example 3. Simplify the expression $\frac{12^4 \times 9^3 \times 4}{6^3 \times 18^2 \times 8}$

Answer:

To simplify such expressions, we first express all bases as powers of their prime factors.

Prime factors of the bases are: 12 = $2^2 \times 3$ 9 = $3^2$ 4 = $2^2$ 6 = $2 \times 3$ 18 = $2 \times 3^2$ 8 = $2^3$

Now substitute these into the expression:

$ \frac{(2^2 \times 3)^4 \times (3^2)^3 \times 2^2}{(2 \times 3)^3 \times (2 \times 3^2)^2 \times 2^3} $

Using Law 4 ($(ab)^m = a^m b^m$) and Law 3 ($(a^m)^n = a^{mn}$):

$ (2^2 \times 3)^4 = (2^2)^4 \times 3^4 = 2^{2 \times 4} \times 3^4 = 2^8 \times 3^4 $

$ (3^2)^3 = 3^{2 \times 3} = 3^6 $

$(2 \times 3)^3 = 2^3 \times 3^3$

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

Substitute these back into the main expression:

$ \frac{(2^8 \times 3^4) \times 3^6 \times 2^2}{(2^3 \times 3^3) \times (2^2 \times 3^4) \times 2^3} $

Group the terms with the same base in the numerator and denominator using Law 1:

Numerator:

$2^8 \times 3^4 \times 3^6 \times 2^2 = (2^8 \times 2^2) \times (3^4 \times 3^6) = 2^{8+2} \times 3^{4+6} = 2^{10} \times 3^{10}$

Denominator:

$2^3 \times 3^3 \times 2^2 \times 3^4 \times 2^3 = (2^3 \times 2^2 \times 2^3) \times (3^3 \times 3^4) = 2^{3+2+3} \times 3^{3+4} = 2^8 \times 3^7$

So the expression becomes:

$ \frac{2^{10} \times 3^{10}}{2^8 \times 3^7} $

Now, separate the terms and use Law 2 ($a^m \div a^n = a^{m-n}$):

$ = \left(\frac{2^{10}}{2^8}\right) \times \left(\frac{3^{10}}{3^7}\right) $

$ = 2^{10-8} \times 3^{10-7} $

$ = 2^2 \times 3^3 $

This is the simplified form in exponential notation. We can also calculate the value:

$ 2^2 \times 3^3 = (2 \times 2) \times (3 \times 3 \times 3) = 4 \times 27 $

$ 4 \times 27 = 108 $

So, the simplified value is 108.

Uses of Exponents

Exponents are not just a mathematical shortcut for writing repeated multiplication; they have significant practical uses, especially when dealing with very large or very small numbers. These numbers are frequently encountered in science, technology, astronomy, and even everyday contexts like population figures or distances. Exponents provide a convenient way to represent such numbers in a compact, readable, and easy-to-compare format, known as Standard Form or Scientific Notation.

Standard Form (Scientific Notation)

The Standard Form or Scientific Notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It is widely used by scientists, engineers, and mathematicians.

A number is said to be in standard form if it is expressed as the product of a number between 1 (inclusive) and 10 (exclusive) and a power of 10.

Standard Form $= k \times 10^n$

Where:

$k$ is a number such that $1 \leq k < 10$. This means $k$ is greater than or equal to 1 but strictly less than 10. It has only one non-zero digit to the left of the decimal point.
$n$ is an integer (it can be a positive whole number, a negative integer, or zero). The integer $n$ is called the exponent of 10.

Writing Large Numbers in Standard Form:

To convert a large number (greater than or equal to 10) into standard form, follow these steps:

Locate the decimal point. In a whole number, the decimal point is assumed to be at the very end (to the right of the last digit).
Move the decimal point to the left until there is only one non-zero digit remaining to the left of the decimal point.
The number obtained in step 2 is the value of $k$.
Count the number of places you moved the decimal point. This count is the exponent $n$. Since you moved the decimal point to the left, the exponent $n$ will be a positive integer.
Write the number in the form $k \times 10^n$.

Example: Write 384,000,000 in standard form.

Start with the number: 384,000,000. The decimal point is initially after the last 0 (384,000,000.).
Move the decimal point to the left until there is only one non-zero digit (which is 3) to the left of the decimal point: 3.84000000.
The number $k$ is 3.84 (we can drop the trailing zeros after the decimal).
Count the number of places the decimal point was moved: $384,000,000.$ $38,400,000.$ (1 place) $3,840,000.$ (2 places) $384,000.$ (3 places) $38,400.$ (4 places) $3,840.$ (5 places) $384.$ (6 places) $38.4$ (7 places) $3.84$ (8 places) We moved the decimal point 8 places to the left. So, $n = 8$.
Write the number in standard form: $3.84 \times 10^8$.

Writing Small Numbers in Standard Form:

To convert a very small number (between 0 and 1) into standard form, follow these steps:

Locate the decimal point.
Move the decimal point to the right until there is only one non-zero digit remaining to the left of the decimal point.
The number obtained in step 2 is the value of $k$.
Count the number of places you moved the decimal point. This count is the exponent $n$. Since you moved the decimal point to the right, the exponent $n$ will be a negative integer.
Write the number in the form $k \times 10^n$.

Example: Write 0.0000056 in standard form.

Start with the number: 0.0000056.
Move the decimal point to the right until there is only one non-zero digit (which is 5) to the left of the decimal point: 5.6.
The number $k$ is 5.6.
Count the number of places the decimal point was moved: $0\textbf{.}0000056$ $00\textbf{.}000056$ (1 place right) $000\textbf{.}00056$ (2 places right) $0000\textbf{.}0056$ (3 places right) $00000\textbf{.}056$ (4 places right) $000000\textbf{.}56$ (5 places right) $0000005\textbf{.}6$ (6 places right) We moved the decimal point 6 places to the right. So, $n = -6$.
Write the number in standard form: $5.6 \times 10^{-6}$.

Note: Writing small numbers in standard form using negative exponents of 10 is typically covered in more detail in higher classes, but understanding how the power of 10 relates to the movement of the decimal point is useful even in Class 7.

Comparing Large Numbers in Standard Form

One of the major advantages of using standard form is that it makes comparing very large (or very small) numbers simple and quick. To compare two numbers written in standard form ($k_1 \times 10^{n_1}$ and $k_2 \times 10^{n_2}$), follow these steps:

Compare the exponents of 10 ($n_1$ and $n_2$). The number with the larger exponent is the greater number. For example, $10^9$ is much larger than $10^8$.
If the exponents of 10 are the same ($n_1 = n_2$), then compare the numbers $k_1$ and $k_2$ (the numbers between 1 and 10). The number with the larger $k$ value is the greater number.

Examples of Comparing Numbers in Standard Form:

Example 1: Compare $1.5 \times 10^9$ and $2.1 \times 10^8$.

Compare the exponents of 10: $10^9$ and $10^8$.
Here, the exponents are 9 and 8.
Since $9 > 8$, we know that $10^9 > 10^8$.
Therefore, $1.5 \times 10^9$ is greater than $2.1 \times 10^8$. ($1.5 \times 10^9 > 2.1 \times 10^8$).

Example 2: Compare $4.7 \times 10^6$ and $3.9 \times 10^6$.

Compare the exponents of 10: $10^6$ and $10^6$.
Here, the exponents are the same (both are 6).
Now, compare the numbers between 1 and 10: 4.7 and 3.9.
Since $4.7 > 3.9$.
Therefore, $4.7 \times 10^6$ is greater than $3.9 \times 10^6$. ($4.7 \times 10^6 > 3.9 \times 10^6$).

Comparing numbers in standard form eliminates the need to write out long strings of zeros, making large numbers manageable and their comparison straightforward.

Example 1. Write the following numbers in standard form:

(a) 5000000

(b) 78900

Answer:

(a) 5000000. The decimal point is after the last 0. We need to move it to the left until it is after the digit 5, so there is only one non-zero digit (5) before the decimal.

5000000. $\xrightarrow{\text{Move decimal left}}$ 5.000000

We moved the decimal point 6 places to the left. So, $n=6$. The number $k=5.0 = 5$.

Standard form: $5 \times 10^6$

(b) 78900. The decimal point is after the last 0. We need to move it to the left until it is after the digit 7.

78900. $\xrightarrow{\text{Move decimal left}}$ 7.8900

We moved the decimal point 4 places to the left. So, $n=4$. The number $k=7.89$ (we can drop trailing zeros).

Standard form: $7.89 \times 10^4$

The decimal point is after the last 0. Move it to the left until it is after the digit 2.

250000. $\xrightarrow{\text{Move decimal left}}$ 2.50000

We moved the decimal point 5 places to the left. So, $n=5$. The number $k=2.5$ (we can drop trailing zeros).

Standard form: $2.5 \times 10^5$

So, the population of the city is approximately $2.5 \times 10^5$.

Example 2. Write the following numbers from standard form to usual form:

(a) $2.5 \times 10^4$

(b) $9.01 \times 10^6$

Answer:

To convert a number from standard form ($k \times 10^n$) to usual form, we move the decimal point of $k$ by $n$ places. If $n$ is positive, we move the decimal point to the right. If $n$ is negative, we move it to the left (this case is for smaller numbers, usually covered later).

(a) $2.5 \times 10^4$. The exponent is 4 (positive), so move the decimal point 4 places to the right from its current position in 2.5.

Current position is between 2 and 5. We need to move 4 places. We have one digit (5) after the decimal, so we need 3 more places. We fill these places with zeros.

$2\textbf{.}5 \xrightarrow{1} 25\textbf{.} \xrightarrow{1} 250\textbf{.} \xrightarrow{1} 2500\textbf{.} \xrightarrow{1} 25000\textbf{.} $

So, $2.5 \times 10^4 = 25000$.

(b) $9.01 \times 10^6$. The exponent is 6 (positive), so move the decimal point 6 places to the right from its current position in 9.01.

Current position is between 9 and 0. We need to move 6 places. We have two digits (01) after the decimal, so we need $6 - 2 = 4$ more places. We fill these places with zeros.

$9\textbf{.}01 \xrightarrow{2 \text{ places}} 901\textbf{.} \xrightarrow{4 \text{ places}} 9010000\textbf{.} $

So, $9.01 \times 10^6 = 9010000$.

(c) $1.005 \times 10^3$. The exponent is 3 (positive), so move the decimal point 3 places to the right from its current position in 1.005.

Current position is between 1 and 0. We need to move 3 places. We have three digits (005) after the decimal, so we need $3 - 3 = 0$ more places (no zeros needed). We just move the decimal point past the last digit 5.

$1\textbf{.}005 \xrightarrow{3 \text{ places}} 1005\textbf{.} $

So, $1.005 \times 10^3 = 1005$.