Cubes and Cube Roots: Properties and Examples

The cube of a number is that number raised to power \(3\). Thus, if \(x\) is a number, then the cube of \(x\) is \(x^3\). The cube root of a number \(x\) is that number whose cube gives \(x\). We denote the cube root of \(x\) by \(\sqrt[3]{x}\). Cubes and Cube Roots are commonly found in mathematical equations

This article will study the definitions of cubes and cube roots of natural numbers, perfect cube numbers, cubes of negative integers, the cube of a rational number, cube root of a number by using its one’s place, and cube root of a perfect cube by factors and so on.

What are Cubes?

Suppose a number is raised to power \(3\). Then, we call it a cube of that number. In other words, if a number multiplies itself \(3\) times, then we call it a cube of that number.
Example: \(2^3 = 2 × 2 × 2 = 8\)
\(⇒\) The cube of \(2\) is \(8\).

Cubes of Natural Numbers

The cube of a natural number is that natural number raised to power \(3\).
Thus, if \(n\) is a natural number, then, \(n^3\) is the cube of \(n\), and we have
\(n^3 = n × n × n\)

Perfect Cube Numbers

A number is a perfect cube if it is a cube of some natural number.
Example: \(27\) is a perfect cube because there is a natural number \(3\) such that \(27 = 3 × 3 × 3\). But, \(12\) is not a perfect cube because there is no natural number whose cube is \(12\).

There are only \(10\) perfect cubes from \(1\) to \(1000\). They are listed below.

Properties of Cubes of Natural Numbers

1. Cubes of all even natural numbers are even.
2. Cubes of all odd numbers are odd.
3. The sum of the cubes of first \(n\) natural numbers is equal to the square of their sum.
   That is, \(1^3 + 2^3 + 3^3 +……+ n^3 = (1 + 2 + 3 +……+ n)^2\)
4. Cubes of the numbers ending in digits \(4, 5, 6\) and \(9\) are the numbers ending in the same digit. Cubes of numbers ending in digit \(2\) ends in digit \(8\), and the cube of numbers ending in digit \(8\) ends in digit \(2\). The cubes of the numbers ending in digits \(3\) and \(7\) ends in digits \(7\) and \(3\), respectively.
5. If a prime number \(p\) divides \(m\), then \(p × p × p\) will divide \(m × m × m\), i.e., \(p^3\) will divide \(m^3\). Thus, if a prime number \(p\) divides a perfect cube, then \(p^3\) also divides this perfect cube.

Finding Cube of a Two-Digit Number by Column Method

This method is based upon an old Indian practice of multiplying two numbers. It is convenient for finding squares of two-digit numbers only. As the number of digits increases, this method becomes difficult and time-consuming. So, we shall discuss it for two-digit numbers only.

This method uses the identity \((a + b)^3 = a^3 + b^3 + 3a^2 b + 3ab^2\) for finding the cube of the two-digit number \(ab\) where \(a\) is the tens digit and \(b\) is the units digit, be a two-digit natural number.

Example: Find a cube of \(42\) by using the column method.
Solution: Let \(a = 4\) and \(b = 2\)
Step 1: Make four columns and write the values of \(a^3, 3a^2 b, 3ab^2\) and \(b^3\) respectively in these columns.
Step 2: Underline the unit digits of \(b^3\) and add its tens digit, if any, to column III.
Step 3: Underline the unit digits of \(3ab^2\) and add its tens digit, if any, to column II
Step 4: Underline the unit digits of \(3a^2 b\) and add its tens digit, if any, to column I.
Step 5: Underline the number in column I.
Step 6: Write the underlined digits at the bottom of each column to obtain the cube of the given number.

Therefore, \(42^3 = 74088\)

Cubes of Negative Integers

The cube of a negative integer is always negative. In general, if \(m\) is a positive integer, then:
\((-m)^3 = -m × -m × -m\)
Example: \((-2)^3 = -2 × -2 × -2 = -8\)

Cubes of Rational Number

Let \(a = \frac{m}{n}\) be a rational number, where \(m\) and \(n\) are non-zero integers, then a cube of \(a\) is defined as \(a^3 = a × a × a\) or
\({\left({\frac{m}{n}} \right)^3} = \frac{m}{n} \times \frac{m}{n} \times \frac{m}{n} = \frac{{{m^3}}}{{{n^3}}}\)

Cube Roots

A number \(m\) is the cube root of a number \(n\) if \(n = m^3\). In other words, the cube root of a number \(n\) is that number \(m\) whose cube gives \(n\).
The cube root of a number \(n\) is denoted by \(\sqrt[3]{n}\).\(\sqrt[3]{n}\) is also called a radical, \(n\) is called the radicand, and \(3\) is called the radical index.

The below table provides all the cube roots upto \(1000\).

Cube Roots Formula

The cube root formula is used to find the cube root of any number stated in radical form with the sign \(∛\).
For example, \(3\) is the number. Then the cube root will be
(3)= (∛3)= (∛(3\times3\times3))= (27)

Cube Root of Natural Numbers

A natural number \(m\) is the cube root of a natural number \(n\) if \(n = m^3\) and we write it as \(\sqrt[3]{n} = m\)

Cube Root of a Number Using Its One’s Place

The units digit of a cube of a natural number depends upon the unit digit of the given number.
The following are the rules to be followed, they are;

1. Look at the digit at the units place and determine the digit at the units place in the cube root.
2. Make groups of three digits, beginning with an extreme right digit of the given number.
   For example, we can group \(15625\) as \(\underline {15} \,\underline {625} \).
   The first group will be \(\underline {625} \) and the second group will be \(\underline {15} \).
3. See the digit at the unit’s place in the first group of three digits in the given number and then identify the digit at the unit’s place in the given number’s cube root using the following table.

So, the unit place of \(15625\) will be \(5\) as the unit place in the first group of three digits of \(15625\) is \(5\).

1. Strikeout units, tens and hundreds places digits. If no digits are left out, then the digit obtained in step \(2\) is the required cube root. If digit/digits are left out, proceed further.
2. Consider the number left after striking out the three digits in step \(2\). Identify the largest one-digit number whose cube is less than or equal to this left-out number. This digit will be the tens digit of the required cube root.
   The number left out in \(15625\) is \(\underline {15} \).
   The largest one-digit number whose cube is less than or equal to \(12\) is \(2^3 = 8\).
3. Now write down the number whose units digit is the number obtained in step \(1\) and the tens digit is the number obtained in step \(3\), then write the required cube root.
   So, \(\sqrt[3]{{15625}} = 25\)

Example: Let us find the cube root of \(64\) by the method mentioned above.
Answer: In the given number \(64\), the unit’s place is \(4\). Therefore, the digit at the one’s place in the cube root is \(4\). Since no number is left after striking out the units and ten’s digit of the number. Therefore, the required cube root is \(4\). i.e. \(\sqrt[3]{{64}} = 4\)

Cube Root of a Perfect Cube by Factors

We use the below steps to find the cube root of a perfect cube by factors.

1. Obtain the given number
2. Resolve it into prime factors
3. Group the factors in triples such that all the three factors in each triple are equal.
4. Take one factor from each triplet formed in the third step.
5. Find the product of factors obtained in the above step. This product is the required cube root.

Cube Root of Product of Integers

For any two integers \(a\) and \(b\), we have \(\sqrt[3]{{ab}} = \sqrt[3]{{a}} \times \sqrt[3]{{b}}\)

Cube Root of a Rational Number

If \(x\) and \(a\) are two rational numbers such that \(x^3 = a\), then we say that \(x\) is the cube root of \(a\) and we write it as \(\sqrt[3]{{a}} = x\).
In other words, for any rational number \(\frac {a}{b}\), we have \(\sqrt[3]{{\frac{a}{b}}} = \frac{{\sqrt[3]{a}}}{{\sqrt[3]{b}}}\)
Example: \({\left({\frac{3}{4}}\right)^3} = \frac{{27}}{{64}}\)
Therefore, \(\sqrt[3]{{\frac{{27}}{{64}}}} = \frac{3}{4}\)

Solved Examples

Q.1. Is \(216\) a perfect cube? What is the number whose cube is \(216\)?
Ans: Resolving \(216\) into prime factors, we get
\(216 = 2 × 2 × 2 × 3 × 3 × 3\)
Grouping the factors in triplets of equal factors, we get
\(216 = [2 × 2 × 2] × [3 × 3 × 3]\)
We find that we can group the prime factors of \(216\) into triples of equal factors, and no factors are leftover.
Therefore, \(216\) is a perfect cube.
Taking one factor from each triplet, we get \(2 × 3 = 6\)
Hence, \(216\) is the cube of \(6\).

Q.2. Using the column method, find the cube of \(87\).
Ans: Let \(a = 8\) and \(b = 7\)
Using the column method, we have

Therefore, \(87^3 = 658503\)

Q.3. Show that \(-17576\) is a perfect cube. Also, find the number whose cube is \(-17576\).
Ans: Resolving \(17576\) into prime factors, we get
\(17576 = 2 × 2 × 2 × 13 × 13 × 13\)
We can group \(17576\) into triples of equal factors, and no factor is left over.
So, \(17576\) is a perfect cube.
Thus, \(-17576\) is also a perfect cube.
Taking one factor from each group, we find that \(17576\) is a perfect cube of \(2 × 13 = 26\)
Hence, \(-17576\) is a perfect cube of \(-26\).

Q.4. Find the cube root \(389017\).
Ans: The given number is \(389017\)
The unit digit of the given number is \(7\). Therefore, the units digit of its cube root is also \(3\).
After striking out the last three-digit from the right, the number left is \(389\).
Now, \(7^3 = 343 < 389\) and \(8^3 = 512 > 389\)
Therefore, the tens digit of the cube root of the given number is \(7\).
Hence, \(\sqrt[3]{{389017}} = 73\)

Q.5. Find the cube root of \(91125\).
Ans: Resolving the given number into prime factors, we get
\(91125 = 5 × 5 × 5 × 3 × 3 × 3 × 3 × 3 × 3\)
Grouping the factors in triples of equal factors, we get
\(91125 = [5 × 5 × 5] × [3 × 3 × 3] × [3 × 3 × 3]\)
Taking one factor from each triple, we get
\(\sqrt[3]{{91125}} = 5 \times 3 \times 3 = 45\)

Summary

In the above article, we have studied the definitions of cubes and cube roots of natural numbers, perfect cube numbers, cubes of negative integers, the cube of a rational number, cube root of a number by using its one’s place and cube root of a perfect cube by factors.

FAQs

Q.1. What are cubes and cube roots?
Ans: The cube of a number is that number raised to power \(3\). Thus, if \(x\) is a number, then the cube of \(x\) is \(x^3\). A number \(m\) is the cube root of a number \(n\) if \(n = m^3\). In other words, the cube root of a number \(n\) is that number \(m\) whose cube gives \(n\).

Q.2. What is the formula for cubes and cube roots?
Ans: Cube of a number \(n = n × n × n\)
For example, let us take the number \(3\). We know that, \(3 × 3 × 3 = 27\). Hence, \(27\) is called the cube of \(3\).
The cube root of a number is the reverse process of the cube of a number. If \(m\) is the cube of \(n\), then \(n = \sqrt[3]{m}\)
For example: \(\sqrt[3]{125} = 5\)

Q.3. How to solve square roots and cube roots?
Ans: Steps to find the cube root of a perfect cube by factors.
1. Obtain the given number and resolve it into prime factors.
2. Group the factors in triples such that all the three factors in each triple are equal and take one factor from each triple formed. Find the product of factors obtained in the above step. This product is the required cube root.
3. Group the factors in doubles such that all the two factors in each double are equal and take one factor from each double formed. Find the product of factors obtained in the above step. This product is the required square root.

Q.4. What is the difference between cubes and cube roots?
Ans: A cube root is a value that gives us the cube number when we multiply it three times. Thus, a perfect cube is a cube of a whole number. A cube root is when we multiply the lowest number three times to arrive at the number.
Example: \(3^3 = 3 × 3 × 3 = 27\) and \(\sqrt[3]{{27}} = 3\)

Q.5. Is \(45\) a perfect cube?
Ans: The prime factors of \(45\) are \(3 × 3 × 5\). Here we do not find any prime number repeated thrice. So, \(45\) is not a perfect cube.