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| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
Chapter 7 Cubes and Cube Roots (Concepts)
Having explored the two-dimensional concept of squaring numbers and the process of finding their roots, we now elevate our understanding to the third dimension by introducing the parallel concepts of Cubes and Cube Roots. Just as squaring involves multiplying a number by itself once ($n \times n$), cubing involves multiplying a number by itself twice ($n \times n \times n$). This chapter delves into this operation, examines the properties of the resulting numbers (perfect cubes), and develops the methodology for performing the inverse operation – extracting the cube root. This extension of numerical operations is fundamental for understanding volume, certain algebraic forms, and various scientific principles.
The cube of a number is formally defined as the result obtained when that number is used as a factor three times in a multiplication. Mathematically, if a number is represented by $n$, its cube is denoted as $n^3$, signifying $n \times n \times n$. For instance, the cube of $4$ is $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$. Similarly, the cube of $10$ is $10^3 = 10 \times 10 \times 10 = 1000$. We will practice calculating the cubes of various numbers, particularly integers. Integers resulting from cubing other integers (like $1, 8, 27, 64, 125, \dots$) are known as perfect cubes.
Perfect cubes possess distinct properties that are helpful for identification and understanding:
- Ending Digits Pattern: The last digit of a perfect cube is determined by the last digit of the number being cubed. There's a specific pattern:
- Numbers ending in 0, 1, 4, 5, 6, 9 have cubes ending in the same digit.
- Numbers ending in 2 have cubes ending in 8.
- Numbers ending in 8 have cubes ending in 2.
- Numbers ending in 3 have cubes ending in 7.
- Numbers ending in 7 have cubes ending in 3.
- Even/Odd Property: The cube of an even number is always even ($6^3 = 216$), and the cube of an odd number is always odd ($5^3 = 125$).
- Sign Property: Unlike squares, where the square of a negative number is positive, the cube of a negative number remains negative. For example, $(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8$. This is a crucial difference from squares.
Occasionally, interesting numerical patterns related to cubes might be mentioned, such as the famous story involving the Hardy-Ramanujan number $1729$, which is the smallest positive integer expressible as the sum of two cubes in two different ways ($1729 = 1^3 + 12^3 = 9^3 + 10^3$).
The inverse operation to cubing is finding the cube root, denoted by the symbol $\sqrt[3]{\phantom{x}}$. The cube root of a given number $x$ is defined as the number $y$ which, when cubed, yields $x$ (i.e., $y^3 = x$). For example, since $6^3 = 216$, the cube root of $216$ is $6$, written as $\sqrt[3]{216} = 6$. A significant distinction from square roots is that every real number (whether positive, negative, or zero) has exactly one real cube root. For instance, $\sqrt[3]{-8} = -2$, because $(-2)^3 = -8$. There isn't a positive counterpart like with square roots.
The primary method taught at this level for determining the cube root, particularly for numbers that are perfect cubes, is the prime factorization method. The steps are as follows:
- Find the prime factorization of the given number.
- Group the identical prime factors into triplets (sets of three).
- For each triplet of a prime factor found, take one instance of that factor out of the cube root operation.
- Multiply these extracted factors together to get the cube root.
For example, to find $\sqrt[3]{1728}$: $1728 = \ $$ 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3)$. Grouping into triplets: $\sqrt[3]{(2^3) \times (2^3) \times (3^3)}$. Extracting one factor for each triplet: $2 \times 2 \times 3 = 12$. Thus, $\sqrt[3]{1728} = 12$. Finding cube roots of large non-perfect cubes usually involves estimation techniques or the use of calculators, as a simple algorithmic method like the long division for square roots is not typically covered. Practical applications often arise in geometry, specifically related to volume calculations, such as finding the length of the edge of a cube when its volume is known ($edge = \sqrt[3]{Volume}$). This chapter serves to further broaden numerical understanding by exploring powers of 3 and their corresponding inverse operation.
Cube Numbers and Perfect Cubes
In the previous chapter, we studied squares and square roots, where squaring a number meant multiplying it by itself ($n \times n$). The geometric connection was the area of a square. Now, we move to cubes and cube roots. The geometric connection for a cube of a number is the volume of a cube.
Consider a cube (a 3D solid with all edges equal in length). If the edge length of a cube is 1 unit, its volume is $1 \times 1 \times 1 = 1$ cubic unit. If the edge length is 2 units, its volume is $2 \times 2 \times 2 = 8$ cubic units. If the edge length is 3 units, its volume is $3 \times 3 \times 3 = 27$ cubic units, and so on.
The numbers 1, 8, 27, 64, 125, ... are obtained when we multiply a number by itself three times.
Definition of Cube Number or Perfect Cube
When a number is multiplied by itself three times, the resulting product is called the cube of that number.
For example, the cube of 4 is $4 \times 4 \times 4 = 64$.
We use a special notation to represent the cube of a number. We write the number and put a small '3' as a superscript above and to the right of it. This '3' indicates that the number is multiplied by itself three times (raised to the power of 3).
The cube of a number $n$ is written as $n^3 = n \times n \times n$.
... (i)
For example:
- Cube of 1 is $1^3 = 1 \times 1 \times 1 = 1$.
- Cube of 2 is $2^3 = 2 \times 2 \times 2 = 8$.
- Cube of 3 is $3^3 = 3 \times 3 \times 3 = 27$.
- Cube of 10 is $10^3 = 10 \times 10 \times 10 = 1000$.
- Cube of 5 is $5^3 = 5 \times 5 \times 5 = 125$.
A natural number (or a positive integer) is called a cube number or a perfect cube if it is the cube of some natural number.
In other words, a natural number $M$ is a perfect cube if there exists a natural number $n$ such that $M = n^3$.
Examples of perfect cubes:
- 1 is a perfect cube because $1 = 1^3$.
- 8 is a perfect cube because $8 = 2^3$.
- 27 is a perfect cube because $27 = 3^3$.
- 216 is a perfect cube because $216 = 6^3$.
- 729 is a perfect cube because $729 = 9^3$.
- 8000 is a perfect cube because $8000 = 20^3$.
Numbers like 2, 3, 4, 5, 6, 7, 9, 10, 11, ... (that are not in the list of cubes) are not perfect cubes because they cannot be expressed as the cube of any natural number.
List of First Few Cube Numbers:
It's useful to know the cubes of the first few natural numbers. Here is a table showing the cubes of natural numbers from 1 to 20:
| Number (n) | Cube ($n^3$) |
|---|---|
| 1 | $1^3 = 1 \times 1 \times 1 = 1$ |
| 2 | $2^3 = 2 \times 2 \times 2 = 8$ |
| 3 | $3^3 = 3 \times 3 \times 3 = 27$ |
| 4 | $4^3 = 4 \times 4 \times 4 = 64$ |
| 5 | $5^3 = 5 \times 5 \times 5 = 125$ |
| 6 | $6^3 = 6 \times 6 \times 6 = 216$ |
| 7 | $7^3 = 7 \times 7 \times 7 = 343$ |
| 8 | $8^3 = 8 \times 8 \times 8 = 512$ |
| 9 | $9^3 = 9 \times 9 \times 9 = 729$ |
| 10 | $10^3 = 10 \times 10 \times 10 = 1000$ |
| 11 | $11^3 = 11 \times 11 \times 11 = 1331$ |
| 12 | $12^3 = 12 \times 12 \times 12 = 1728$ |
| 13 | $13^3 = 13 \times 13 \times 13 = 2197$ |
| 14 | $14^3 = 14 \times 14 \times 14 = 2744$ |
| 15 | $15^3 = 15 \times 15 \times 15 = 3375$ |
| 16 | $16^3 = 16 \times 16 \times 16 = 4096$ |
| 17 | $17^3 = 17 \times 17 \times 17 = 4913$ |
| 18 | $18^3 = 18 \times 18 \times 18 = 5832$ |
| 19 | $19^3 = 19 \times 19 \times 19 = 6859$ |
| 20 | $20^3 = 20 \times 20 \times 20 = 8000$ |
You can see that cube numbers grow much faster than square numbers.
Checking if a Number is a Perfect Cube
Similar to perfect squares, not every natural number is a perfect cube. To check if a number is a perfect cube, we need to see if it can be expressed as the product of three equal natural numbers. We will learn a systematic method (prime factorisation) later in this chapter.
For now, simply comparing a number to the list of known cubes can help identify if it's a perfect cube, especially for smaller numbers.
Example 1. Is 200 a perfect cube?
Answer:
We need to check if there is a natural number $n$ such that $n^3 = 200$.
Let's look at the cubes of natural numbers:
$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
$5^3 = 125$
$6^3 = 216$
We see that $5^3 = 125$, which is less than 200, and $6^3 = 216$, which is greater than 200.
There is no natural number between 5 and 6 whose cube would be exactly 200.
Therefore, 200 is not a perfect cube.
Properties of Cubes of Natural Numbers
We have defined cube numbers and listed the cubes of the first few natural numbers. Let's investigate some properties of these cube numbers. Observing these properties can help us understand the characteristics of perfect cubes and sometimes make predictions or checks without full calculation.
Property 1: Unit Digits of Cube Numbers
The unit digit of the cube of a natural number is determined solely by the unit digit of the original number. Let's look at the unit digits of the cubes from our previous list:
- $1^3 = 1$ (Unit digit: 1)
- $2^3 = 8$ (Unit digit: 8)
- $3^3 = 27$ (Unit digit: 7)
- $4^3 = 64$ (Unit digit: 4)
- $5^3 = 125$ (Unit digit: 5)
- $6^3 = 216$ (Unit digit: 6)
- $7^3 = 343$ (Unit digit: 3)
- $8^3 = 512$ (Unit digit: 2)
- $9^3 = 729$ (Unit digit: 9)
- $10^3 = 1000$ (Unit digit: 0)
Unlike square numbers (which only end in 0, 1, 4, 5, 6, 9), the unit digits of perfect cubes can be any digit from 0 to 9. This means looking at the unit digit alone cannot tell us if a number is *not* a perfect cube, except in combination with other properties (like the number of zeros).
Let's summarise the mapping of unit digits of a number to the unit digit of its cube:
- If a number ends in 0, its cube ends in 0.
- If a number ends in 1, its cube ends in 1.
- If a number ends in 2, its cube ends in 8.
- If a number ends in 3, its cube ends in 7.
- If a number ends in 4, its cube ends in 4.
- If a number ends in 5, its cube ends in 5.
- If a number ends in 6, its cube ends in 6.
- If a number ends in 7, its cube ends in 3.
- If a number ends in 8, its cube ends in 2.
- If a number ends in 9, its cube ends in 9.
Notice an interesting pairing: 2 and 8 are swapped (2 maps to 8, 8 maps to 2), and 3 and 7 are swapped (3 maps to 7, 7 maps to 3). For all other digits (0, 1, 4, 5, 6, 9), the unit digit of the cube is the same as the unit digit of the original number.
This property is useful for determining the unit digit of a cube without cubing the entire number, and also for estimating the unit digit of a cube root (discussed later).
Example 1. What is the unit digit of the cube of 73?
Answer:
Given number: 73.
The unit digit of the number 73 is 3.
According to Property 1, if a number ends in 3, its cube ends in 7.
Therefore, the unit digit of the cube of 73 (i.e., $73^3$) is 7.
Check: $73 \times 73 = 5329$. The unit digit of 5329 is 9. $73^3 = 5329 \times 73$. The unit digit of the product is the unit digit of $9 \times 3 = 27$, which is 7. The property holds.
Property 2: Cubes of Even and Odd Numbers
Let's look at the cubes of some even and odd natural numbers:
- Even numbers: $2^3=8$ (even), $4^3=64$ (even), $6^3=216$ (even), $10^3=1000$ (even), $12^3=1728$ (even).
- Odd numbers: $1^3=1$ (odd), $3^3=27$ (odd), $5^3=125$ (odd), $7^3=343$ (odd), $9^3=729$ (odd).
Observation: The cube of an even number is always an even number. The cube of an odd number is always an odd number.
This property is useful: if a perfect cube is even, its cube root must be even. If a perfect cube is odd, its cube root must be odd.
Property 3: Cubes of Numbers Ending in 0
Consider the cubes of numbers that end in zero:
- $10^3 = 10 \times 10 \times 10 = 1000$ (ends in 3 zeros)
- $20^3 = 20 \times 20 \times 20 = 8000$ (ends in 3 zeros)
- $50^3 = 50 \times 50 \times 50 = 125000$ (ends in 3 zeros)
- $100^3 = 100 \times 100 \times 100 = 1000000$ (ends in 6 zeros)
If a number ends in one zero, its cube ends in three zeros. If a number ends in two zeros, its cube ends in six zeros. In general, if a number ends in $k$ zeros, its cube ends in $3k$ zeros.
Property: A perfect cube must end with a number of zeros that is a multiple of 3 (e.g., 3, 6, 9, ... zeros). A number ending with 1, 2, 4, 5, 7, 8 zeros cannot be a perfect cube.
Example 2. Without finding the cube, can you say which of the following are likely perfect cubes?
27000, 800, 125000, 36000, 1000000, 49000
Answer:
We examine the number of zeros at the end of each number:
- 27000: Ends in 3 zeros (a multiple of 3). This number *might* be a perfect cube. (Indeed, $30^3 = 27000$).
- 800: Ends in 2 zeros (not a multiple of 3). This number is not a perfect cube.
- 125000: Ends in 3 zeros (a multiple of 3). This number *might* be a perfect cube. (Indeed, $50^3 = 125000$).
- 36000: Ends in 3 zeros (a multiple of 3). This number *might* be a perfect cube. (We know $10^3=1000$ and $30^3=27000, 40^3=64000$. $3^3=27, 4^3=64$. The non-zero part 36 is not a perfect cube, so 36000 is not a perfect cube despite having 3 zeros).
- 1000000: Ends in 6 zeros (a multiple of 3). This number *might* be a perfect cube. (Indeed, $100^3 = 1000000$).
- 49000: Ends in 3 zeros (a multiple of 3). This number *might* be a perfect cube. (The non-zero part 49 is not a perfect cube, so 49000 is not a perfect cube).
Based *only* on the number of zeros, 27000, 125000, 36000, 1000000, and 49000 are *potentially* perfect cubes. However, combining with the rule that the non-zero part must also be a perfect cube, the numbers that are likely perfect cubes are 27000 ($27$ is a perfect cube), 125000 ($125$ is a perfect cube), and 1000000 ($1$ is a perfect cube). The numbers 800, 36000, and 49000 are not perfect cubes.
This property, like the unit digit property, helps in quickly identifying numbers that are *not* perfect cubes (if the number of zeros is not a multiple of 3). If the number of zeros *is* a multiple of 3, you need to check the non-zero part.
Cubes of Negative Numbers and Rational Numbers
So far, we have focused on the cubes of natural numbers (positive integers). Let's now extend the concept of cubing a number to include negative integers and rational numbers.
Cube of a Negative Number
Let's find the cube of a negative integer. We multiply the negative number by itself three times.
Consider a negative number, say $-a$, where $a$ is a positive number ($a > 0$).
The cube of $(-a)$ is $(-a)^3 = (-a) \times (-a) \times (-a)$
Let's perform the multiplication step by step:
$(-a) \times (-a) = a^2$
(Product of two negative numbers is positive)
Now multiply the result by the remaining $(-a)$:
$(\text{result from first step}) \times (-a) = a^2 \times (-a)$
Remember that a positive number multiplied by a negative number gives a negative result:
$= - (a^2 \times a) = -a^3$
... (i)
So, the cube of a negative number is a negative number, and its value is the negative of the cube of the corresponding positive number.
Property: The cube of a negative number is always a negative number.
Examples:
- Find the cube of -2. $(-2)^3 = (-2) \times (-2) \times (-2) = (4) \times (-2) = -8$. Note that $-(2^3) = -8$.
- Find the cube of -5. $(-5)^3 = (-5) \times (-5) \times (-5) = (25) \times (-5) = -125$. Note that $-(5^3) = -125$.
- Find the cube of -1. $(-1)^3 = (-1) \times (-1) \times (-1) = (1) \times (-1) = -1$. Note that $-(1^3) = -1$.
Cube of a Rational Number (Fraction)
A rational number can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. To find the cube of a rational number, we multiply the fraction by itself three times.
$(\frac{a}{b})^3 = \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b}$
When multiplying fractions, we multiply the numerators together and the denominators together:
$= \frac{a \times a \times a}{b \times b \times b} = \frac{a^3}{b^3}$
... (ii)
Property: The cube of a fraction $\frac{a}{b}$ is obtained by cubing the numerator ($a$) and cubing the denominator ($b$) separately.
Example 1. Find the cube of $\frac{2}{3}$.
Answer:
Given rational number: $\frac{2}{3}$.
Using the formula $(\frac{a}{b})^3 = \frac{a^3}{b^3}$:
$(\frac{2}{3})^3 = \frac{2^3}{3^3}$
Calculate the cube of the numerator (2) and the denominator (3):
$2^3 = 2 \times 2 \times 2 = 8$
$3^3 = 3 \times 3 \times 3 = 27$
Substitute these values:
$(\frac{2}{3})^3 = \frac{8}{27}$
The cube of $\frac{2}{3}$ is $\frac{8}{27}$.
Example 2. Find the cube of $-\frac{1}{5}$.
Answer:
Given rational number: $-\frac{1}{5}$. This is a negative fraction.
Using the formula $(\frac{a}{b})^3 = \frac{a^3}{b^3}$, where $a = -1$ and $b = 5$:
$(-\frac{1}{5})^3 = \frac{(-1)^3}{5^3}$
Calculate the cube of the numerator (-1) and the denominator (5):
$(-1)^3 = (-1) \times (-1) \times (-1) = 1 \times (-1) = -1$
$5^3 = 5 \times 5 \times 5 = 125$
Substitute these values:
$(-\frac{1}{5})^3 = \frac{-1}{125}$
The cube of $-\frac{1}{5}$ is $\frac{-1}{125}$. This can also be written as $-\frac{1}{125}$.
Cube Roots
We have learned how to find the cube of a number by multiplying it by itself three times. The inverse operation of finding the cube of a number is finding its cube root. Finding the cube root of a number $n$ is like asking: "What number, when multiplied by itself three times, gives $n$?"
Definition of Cube Root
If the cube of a number $m$ is equal to another number $n$, i.e., $m^3 = n$, then $m$ is called the cube root of $n$.
In simpler terms, the cube root of a number $n$ is the number $m$ such that when $m$ is cubed ($m \times m \times m$), you get $n$.
The symbol used to denote the cube root is $\sqrt[3]{}$. This symbol is the radical sign with a small '3' index to indicate the cube root.
So, if $m^3 = n$, we write $\sqrt[3]{n} = m$.
Examples:
- We know $2^3 = 8$. So, the cube root of 8 is 2. We write $\sqrt[3]{8} = 2$.
- We know $4^3 = 64$. So, the cube root of 64 is 4. We write $\sqrt[3]{64} = 4$.
- We know $6^3 = 216$. So, the cube root of 216 is 6. We write $\sqrt[3]{216} = 6$.
- We know $10^3 = 1000$. So, the cube root of 1000 is 10. We write $\sqrt[3]{1000} = 10$.
- We know $15^3 = 3375$. So, the cube root of 3375 is 15. We write $\sqrt[3]{3375} = 15$.
Finding the cube root is looking for the base number when the exponent is 3.
Cube Roots of Positive, Negative, and Zero Numbers
Unlike the square root of a positive number which has two real values (one positive, one negative), every real number (positive, negative, or zero) has exactly one real cube root.
Cube Root of a Positive Number:
The cube root of a positive number is always a positive number.
Example: $\sqrt[3]{27} = 3$ because $3^3 = 27$. $\sqrt[3]{125} = 5$ because $5^3 = 125$. $\sqrt[3]{729} = 9$ because $9^3 = 729$.Cube Root of a Negative Number:
The cube root of a negative number is always a negative number.
Example: $\sqrt[3]{-8} = -2$ because $(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8$. $\sqrt[3]{-125} = -5$ because $(-5)^3 = -125$. $\sqrt[3]{-1000} = -10$ because $(-10)^3 = -1000$. In general, $\sqrt[3]{-n} = -\sqrt[3]{n}$ for any positive number $n$.Cube Root of Zero:
The cube root of zero is zero.
Example: $\sqrt[3]{0} = 0$ because $0^3 = 0$.
In summary, for any real number $n$, there is a unique real number $m$ such that $m^3 = n$, and $m$ is the cube root of $n$.
Example 1. What is the cube root of 64?
Answer:
We are looking for the number $m$ such that $m^3 = 64$.
Let's test the cubes of small natural numbers:
$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
Since $4^3 = 64$, the cube root of 64 is 4.
$\sqrt[3]{64} = 4$
Example 2. Find the cube root of -216.
Answer:
We are looking for the number $m$ such that $m^3 = -216$.
Since the number is negative, its cube root must be negative. Let's find the cube root of the corresponding positive number, 216.
We need a number $n$ such that $n^3 = 216$.
Let's test the cubes of small natural numbers:
$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
$5^3 = 125$
$6^3 = 216$
Since $6^3 = 216$, the cube root of 216 is 6 ($\sqrt[3]{216} = 6$).
Therefore, the cube root of -216 is the negative of the cube root of 216.
$\sqrt[3]{-216} = -\sqrt[3]{216} = -6$
Check: $(-6)^3 = (-6) \times (-6) \times (-6) = 36 \times (-6) = -216$. Correct.
Cube Root through Prime Factorisation Method
Similar to finding square roots, the prime factorisation method is a fundamental way to find the cube root of a perfect cube. The underlying principle is based on the definition of a perfect cube and the property of prime factorisation.
A number is a perfect cube if it can be expressed as the product of three equal factors. In terms of prime factors, this means that in the prime factorisation of a perfect cube, each prime factor must appear a number of times that is a multiple of 3.
Steps for Finding Cube Root by Prime Factorisation:
To find the cube root of a given perfect cube number using prime factorisation, follow these steps:
- Find Prime Factors: Find the prime factorisation of the given number. Express the number as a product of its prime factors raised to their respective powers.
- Group Factors in Triplets: Group the identical prime factors in sets of three (triplets). For a number to be a perfect cube, every prime factor must be part of a complete triplet.
- Take One Factor from Each Triplet: For each triplet of identical prime factors, take only one of the factors from the group.
- Multiply the Selected Factors: Multiply the single factors chosen from each triplet. The product of these factors will be the cube root of the original number.
If, after prime factorisation, you cannot group all prime factors into complete triplets of identical factors (i.e., some factor appears 1, 2, 4, 5, 7, 8, ... times), then the original number is not a perfect cube.
Example 1. Find the cube root of 216 by prime factorisation.
Answer:
We need to find $\sqrt[3]{216}$.
Step 1: Find the prime factorisation of 216.
$\begin{array}{c|cc} 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$
The prime factorisation of 216 is $2 \times 2 \times 2 \times 3 \times 3 \times 3$.
Step 2: Group the identical prime factors in triplets.
$216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)$
We can see that the prime factor 2 forms one triplet, and the prime factor 3 forms one triplet. All factors are grouped into triplets. This confirms that 216 is a perfect cube.
Step 3: Take one factor from each triplet.
From the triplet of 2s, take one 2.
From the triplet of 3s, take one 3.
Step 4: Multiply the selected factors to find the cube root.
$\sqrt[3]{216} = 2 \times 3 = 6$
The cube root of 216 is 6.
Check: $6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$. Correct.
Example 2. Find the cube root of 1728 by prime factorisation.
Answer:
We need to find $\sqrt[3]{1728}$.
Step 1: Find the prime factorisation of 1728.
$\begin{array}{c|cc} 2 & 1728 \\ \hline 2 & 864 \\ \hline 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$
The prime factorisation of 1728 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$.
Step 2: Group the identical prime factors in triplets.
$1728 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) = 2^3 \times 2^3 \times 3^3$
All prime factors (2 and 3) are grouped into triplets. This confirms that 1728 is a perfect cube.
Step 3: Take one factor from each triplet.
From the first triplet of 2s, take one 2.
From the second triplet of 2s, take one 2.
From the triplet of 3s, take one 3.
Step 4: Multiply the selected factors to find the cube root.
$\sqrt[3]{1728} = 2 \times 2 \times 3 = 12$
The cube root of 1728 is 12.
Check: $12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728$. Correct.
Checking if a Number is a Perfect Cube using Prime Factorisation
As mentioned, the prime factorisation method can definitively tell us if a number is a perfect cube. A natural number is a perfect cube if and only if, in its prime factorisation, the exponent of every prime factor is a multiple of 3 (i.e., every prime factor can be grouped into complete triplets).
Example 3. Is 243 a perfect cube? Justify your answer using prime factorisation.
Answer:
We need to determine if 243 is a perfect cube.
Find the prime factorisation of 243.
$\begin{array}{c|cc} 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$
The prime factorisation of 243 is $3 \times 3 \times 3 \times 3 \times 3 = 3^5$.
Group the factors in triplets:
$243 = (3 \times 3 \times 3) \times 3 \times 3 = 3^3 \times 3^2$
In the prime factorisation of 243, the factor 3 appears 5 times. When we group in triplets, we get one triplet of 3s, and two factors of 3 are left over (or the exponent 5 is not a multiple of 3).
Since all prime factors (in this case, only 3) do not form complete triplets, 243 is not a perfect cube.
Finding the Smallest Number to Multiply or Divide to Make a Perfect Cube
If a number is not a perfect cube, we can use its prime factorisation to find the smallest number by which it must be multiplied or divided to make it a perfect cube. We need to identify the factors that are not part of a complete triplet.
Example 4. Find the smallest number by which 392 must be multiplied so that the product is a perfect cube.
Answer:
Given number: 392.
Find the prime factorisation of 392.
$\begin{array}{c|cc} 2 & 392 \\ \hline 2 & 196 \\ \hline 2 & 98 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \end{array}$
The prime factorisation of 392 is $2 \times 2 \times 2 \times 7 \times 7 = 2^3 \times 7^2$.
Group the factors in triplets:
$392 = (2 \times 2 \times 2) \times (7 \times 7)$
The factor 2 forms a complete triplet ($2^3$). The factor 7 appears twice ($7^2$), which is not a complete triplet. To make the factor 7 a complete triplet ($7^3$), we need one more factor of 7.
So, to make 392 a perfect cube, we must multiply it by 7.
The smallest natural number by which 392 must be multiplied is 7.
The resulting perfect cube will be $392 \times 7 = (2^3 \times 7^2) \times 7^1 = 2^3 \times 7^{2+1} = 2^3 \times 7^3 = (2 \times 7)^3 = 14^3$.
Example 5. Find the smallest number by which 135 must be divided so that the quotient is a perfect cube.
Answer:
Given number: 135.
Find the prime factorisation of 135.
$\begin{array}{c|cc} 3 & 135 \\ \hline 3 & 45 \\ \hline 3 & 15 \\ \hline 5 & 5 \\ \hline & 1 \end{array}$
The prime factorisation of 135 is $3 \times 3 \times 3 \times 5 = 3^3 \times 5^1$.
Group the factors in triplets:
$135 = (3 \times 3 \times 3) \times 5$
The factor 3 forms a complete triplet ($3^3$). The factor 5 appears once ($5^1$), which is not part of a complete triplet.
To make the quotient a perfect cube, we must remove the factors that are not part of a complete triplet by dividing by them. The unpaired factor is 5. We need to divide 135 by 5.
The smallest natural number by which 135 must be divided is 5.
The resulting number (quotient) is $135 \div 5 = (3^3 \times 5^1) \div 5^1 = 3^3 = 27$. 27 is a perfect cube ($3^3$).
The cube root of the resulting perfect cube (27) is $\sqrt[3]{27} = 3$.
Cube Root through Estimation
For smaller perfect cubes, particularly those up to 6 digits, we can estimate their cube root using a simple method that leverages the properties of the unit digits of cubes. This method provides a quick way to find the cube root without lengthy calculations like prime factorisation, provided you know that the given number is indeed a perfect cube.
Steps for Estimating Cube Root of a Perfect Cube:
This estimation method works by determining the unit digit and the tens digit of the cube root separately.
- Group the Digits: Starting from the rightmost digit (unit digit) of the given number, make groups of three digits. Place a bar over each group. The leftmost group may have one, two, or three digits. For example, for a 6-digit number, you'll get two groups of three. For a 5-digit number, you'll get a group of two and a group of three. For a 4-digit number, a group of one and a group of three.
- Determine the Unit Digit of the Cube Root: Look at the unit digit of the first group (the rightmost group of three digits). The unit digit of the cube root is determined by the unit digit of the original number's unit digit. Use the mapping from the unit digit of a number to the unit digit of its cube (Property 1):
- If the unit digit is 0, the cube root ends in 0.
- If the unit digit is 1, the cube root ends in 1.
- If the unit digit is 2, the cube root ends in 8.
- If the unit digit is 3, the cube root ends in 7.
- If the unit digit is 4, the cube root ends in 4.
- If the unit digit is 5, the cube root ends in 5.
- If the unit digit is 6, the cube root ends in 6.
- If the unit digit is 7, the cube root ends in 3.
- If the unit digit is 8, the cube root ends in 2.
- If the unit digit is 9, the cube root ends in 9.
- Determine the Tens Digit of the Cube Root: Consider the second group from the right (which is the leftmost group). Find the largest integer whose cube is less than or equal to the number in this leftmost group. This integer will be the tens digit of the cube root.
- Combine the Digits: Combine the tens digit found in Step 3 and the unit digit found in Step 2. This two-digit number is the estimated cube root. (For numbers up to 6 digits, the cube root will be a 1-digit or 2-digit number. This method is most straightforward for perfect cubes with a 2-digit cube root).
Example 1. Estimate the cube root of 17576.
Answer:
Given number: 17576.
Step 1: Make groups of three digits from right to left. Place bars over the groups:
$\overline{17} \ \overline{576}$
The two groups are the first group (from the right) which is 576, and the second group (leftmost) which is 17.
Step 2: Consider the first group, 576. The unit digit of 576 is 6. The unit digit of the cube root of a number ending in 6 is 6 (because $6^3 = 216$ ends in 6). So, the unit digit of $\sqrt[3]{17576}$ is 6.
Step 3: Consider the second group from the right (leftmost group), which is 17. Find the largest integer whose cube is less than or equal to 17.
Let's check the cubes of small integers:
$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
The largest integer whose cube is less than or equal to 17 is 2 (since $2^3 = 8 \leq 17$, but $3^3 = 27 > 17$). So, the tens digit of $\sqrt[3]{17576}$ is 2.
Step 4: Combine the tens digit (2) and the unit digit (6). The estimated cube root is 26.
Since 17576 is a perfect cube, this estimated value is its exact cube root.
So, $\sqrt[3]{17576} = 26$.
Check: $26^3 = 26 \times 26 \times 26 = 676 \times 26 = 17576$. The estimation is correct.
Example 2. Estimate the cube root of 4913.
Answer:
Given number: 4913.
Step 1: Make groups of three digits from right to left. Place bars over the groups:
$\overline{4} \ \overline{913}$
The two groups are the first group (from the right) which is 913, and the second group (leftmost) which is 4.
Step 2: Consider the first group, 913. The unit digit of 913 is 3. The unit digit of the cube root of a number ending in 3 is 7 (because $7^3 = 343$ ends in 3). So, the unit digit of $\sqrt[3]{4913}$ is 7.
Step 3: Consider the second group from the right (leftmost group), which is 4. Find the largest integer whose cube is less than or equal to 4.
Let's check the cubes of small integers:
$1^3 = 1$
$2^3 = 8$
The largest integer whose cube is less than or equal to 4 is 1 (since $1^3 = 1 \leq 4$, but $2^3 = 8 > 4$). So, the tens digit of $\sqrt[3]{4913}$ is 1.
Step 4: Combine the tens digit (1) and the unit digit (7). The estimated cube root is 17.
Since 4913 is a perfect cube, this estimated value is its exact cube root.
So, $\sqrt[3]{4913} = 17$.
Check: $17^3 = 17 \times 17 \times 17 = 289 \times 17 = 4913$. The estimation is correct.
Cube Root of a Negative Number, Product of Integers and Rational Number
We have learned how to find the cube root of positive perfect cubes using prime factorisation and estimation. Now, let's look at how to find the cube roots of negative numbers, products of numbers, and rational numbers.
Cube Root of a Negative Number
As we discussed in Section I4, every negative number has a unique negative real cube root. The cube root of a negative number $-n$ is the negative of the cube root of the corresponding positive number $n$.
Formula:
$\sqrt[3]{-x} = -\sqrt[3]{x}$
[For any positive real number $x$]
Examples:
- Find the cube root of -27. We know $\sqrt[3]{27} = 3$. So, $\sqrt[3]{-27} = -\sqrt[3]{27} = -3$. Check: $(-3)^3 = -27$.
- Find the cube root of -125. We know $\sqrt[3]{125} = 5$. So, $\sqrt[3]{-125} = -\sqrt[3]{125} = -5$. Check: $(-5)^3 = -125$.
- Find the cube root of -1000. We know $\sqrt[3]{1000} = 10$. So, $\sqrt[3]{-1000} = -\sqrt[3]{1000} = -10$. Check: $(-10)^3 = -1000$.
To find the cube root of a negative number, find the cube root of its absolute value (the positive part) and then put a negative sign in front of the result.
Cube Root of a Product of Integers
The cube root of a product of integers is equal to the product of their individual cube roots.
Formula:
$\sqrt[3]{x \times y} = \sqrt[3]{x} \times \sqrt[3]{y}$
[For any real numbers $x, y$]
This property can be extended to the product of three or more integers: $\sqrt[3]{x \times y \times z} = \sqrt[3]{x} \times \sqrt[3]{y} \times \sqrt[3]{z}$, and so on.
This property is directly used in the prime factorisation method, where we group identical prime factors (which are multiplied together) and take one from each group (finding the cube root of that group) and then multiply these results.
Example: Find the cube root of $8 \times 27$.
$\sqrt[3]{8 \times 27} = \sqrt[3]{8} \times \sqrt[3]{27}$
[Using the property]
We know $\sqrt[3]{8} = 2$ and $\sqrt[3]{27} = 3$.
$= 2 \times 3 = 6$
The cube root of $8 \times 27$ is 6.
Check: Calculate the product first: $8 \times 27 = 216$. Now find the cube root of 216: $\sqrt[3]{216} = 6$ (since $6^3=216$). The result matches.
Cube Root of a Rational Number (Fraction)
To find the cube root of a fraction $\frac{a}{b}$, where $b \neq 0$, we find the cube root of the numerator ($a$) and divide it by the cube root of the denominator ($b$).
Formula:
$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$
[Where $\sqrt[3]{a}$ and $\sqrt[3]{b}$ exist as real numbers and $b \neq 0$]
This property holds because $(\frac{\sqrt[3]{a}}{\sqrt[3]{b}})^3 = \frac{(\sqrt[3]{a})^3}{(\sqrt[3]{b})^3} = \frac{a}{b}$.
Example 1. Find the cube root of $\frac{64}{125}$.
Answer:
Given fraction: $\frac{64}{125}$. Both numerator and denominator are perfect cubes ($64=4^3$, $125=5^3$).
Using the formula for the cube root of a fraction:
$\sqrt[3]{\frac{64}{125}} = \frac{\sqrt[3]{64}}{\sqrt[3]{125}}$
[Using the property]
Find the cube root of the numerator and the denominator separately:
$\sqrt[3]{64} = 4$ ($4^3 = 64$)
$\sqrt[3]{125} = 5$ ($5^3 = 125$)
Substitute these values back into the fraction:
$\sqrt[3]{\frac{64}{125}} = \frac{4}{5}$
The cube root of $\frac{64}{125}$ is $\frac{4}{5}$.
Check: Cube the result: $(\frac{4}{5})^3 = \frac{4^3}{5^3} = \frac{64}{125}$. This matches the original fraction, so the answer is correct.
Example 2. Find the cube root of $4\frac{12}{125}$.
Answer:
Given mixed number: $4\frac{12}{125}$.
Step 1: Convert the mixed number to an improper fraction.
$4\frac{12}{125} = \frac{(4 \times 125) + 12}{125}$
$= \frac{500 + 12}{125} = \frac{512}{125}$
Step 2: Find the cube root of the resulting improper fraction $\frac{512}{125}$.
Using the formula $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$:
$\sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}}$
Find the cube root of the numerator and the denominator. We can use prime factorisation if needed, or recall from the list of cubes:
$\sqrt[3]{512} = 8$ ($8^3 = 512$)
$\sqrt[3]{125} = 5$ ($5^3 = 125$)
Substitute the values:
$\sqrt[3]{4\frac{12}{125}} = \frac{8}{5}$
The cube root of $4\frac{12}{125}$ is $\frac{8}{5}$. This can be written as a mixed number $1\frac{3}{5}$ or a decimal 1.6.
Check: Cube the result: $(\frac{8}{5})^3 = \frac{8^3}{5^3} = \frac{512}{125}$. This is equal to $4\frac{12}{125}$, so the answer is correct.