In this chapter we will learn about the concept of perfect cubes along with the method to calculate cube root of given number.

## What is Perfect Cube ?

A perfect cube is a number formed when we **multiply integer thrice by itself**.

Hence, **when a number is cubed, we get a perfect cube.**

**For example;**

Let’s multiply the number 5 by itself three times.

\mathtt{\Longrightarrow \ 5\times 5\times 5}\\\ \\ \mathtt{\Longrightarrow \ 125}

Here number 125 is a perfect cube since it is formed by multiplying number 3 times.

**Example 02**

Multiply number 12 by itself three times.

\mathtt{\Longrightarrow \ 12\times 12\times 12}\\\ \\ \mathtt{\Longrightarrow \ 1728}

Here the number **1728 is the perfect cube.**

## Check if number is perfect cube or not ?

Suppose you are given a number 3375.

To check if the below number is perfect cube or not, **follow the below steps**;

(i) Find **prime factorization** given number

(ii) **Arrange all the factors with exponent 3**

(iii) **If all the factors are arranged with power 3** then the given number is a **perfect cube**.

(iv) Remove all the exponents and multiply the factor to get the cube root.

Let’s take the example of **number 3375**

**(i) Do the prime factorization.**

**(ii) Write all the factors produced in prime factorization**.

\mathtt{3375\ =3^{3} \times 5^{3}}

Note that all the factors have power 3, hence the number 3375 is a perfect cube.

**(iii) Remove all the exponent of three to get the cube root.**

Cube Root 3375 ⟹ 3 x 5

Cube root 3375 ⟹ 15

Hence, if we multiply the number 15 by itself thrice, we get number 3375.

**Example 02**

Check if 512 is perfect cube

**Solution**

Follow the below steps;

**(a) Do the prime factorization**

**(ii) The factorization can be expressed as;**

\mathtt{512\ \Longrightarrow \ 2^{3} \times 2^{3} \times 2^{3}}

In the calculation, try to form a pair of three for every given factor.

Factor 2 ⟹ form pair of three

Factor 2 ⟹ form pair of three

Factor 2 ⟹ form pair of three

Since all the factors are arranged with exponent 3, the number 512 is a perfect cube.

**(iii) Remove all exponent of 3 to get the cube root**

Cube Root 512 ⟹ 2 x 2 x 2

Cube root 512 ⟹ 8

Hence, if we multiply 8 by itself thrice we get 512.

**Example 03**

Check if number 108 is perfect cube

**Solution**

Follow the below steps;

**(a) Do the prime factorization**

The prime factorization is given as;

\mathtt{108\ \Longrightarrow \ 2^{2} \times 3^{3}}

**(b) Check if all factors forming pair of three**.

Factor 2 ⟹ doesn’t form pair of 3

Factor 3 ⟹ form pair of 3

Since the factor doesn’t form pair of three, the **number 108 is not a perfect cube**.

**Example 04**

Check if number 1000 is a perfect cube

**Solution**

Follow the below steps;

**(a) Do the prime factorization**

The above prime factorization is expressed as;

\mathtt{1000\ \Longrightarrow \ 2^{3} \times 5^{3}}

**(ii) Check if all factor form pair of three**

Factor 2 ⟹ form pair of 3

Factor 5 ⟹ form pair of 3

Since all the factors have 3 as an exponent, the number 1000 is a perfect cube.

**(iii) Remove the exponent and multiply the number, you will get the cube root.**

Cube Root 1000 ⟹ 2 x 5

Cube root 1000 ⟹ 10

Hence, if we multiply number 10 by itself thrice, we get 1000.

**Example 05**

Check if number 1715 is perfect cube.

**Solution****(i) Do the prime factorization**

The prime factorization is expressed as;

\mathtt{1715\ \Longrightarrow \ 5\times 7^{3}}

**(ii) Check if all factor form pair of three.**

Factor 5 ⟹ doesn’t form pair of 3

Factor 7 ⟹ form pair of 3

Since the factor 5 doesn’t contain exponent of three, **the number 1715 is not a perfect cube**.