Cube Root


In this post we will learn about the concept of cube root and the method to find cube root of any number.

What is Cube root ?


Let ” a ” be positive integer.

When we take cube of the number, we get \mathtt{( a)^{3}}

Now, if we take the cube root of number \mathtt{( a)^{3}} , we get the original number a.

cube root definition

Hence, cube root is an inverse operation of cubing the number.

How to represent cube root of number ?


If you want to find cube root of any number, place \mathtt{\sqrt[3]{}} sign above that number.

For example;

cube root of 8 ⟹ \mathtt{\sqrt[3]{8}}

cube root of 125 ⟹ \mathtt{\sqrt[3]{125}}

How to find cube root of number ?

Here we will learn to find the cube root of perfect cube.

Follow the below steps;

(a) Do prime factorization of number


(b) Arrange the factors in pair of three.

For perfect cube, we will get all the factors raised to the power of three.


(c) Remove the powers and multiply the factors, you will get the cube of number.

I hope you understood the above process, let us solve some examples for practice.

Example 01
Find cube root of number 216

Solution
Follow the below steps;

(a) Do the prime factorization

cube root examples

The prime factorization can be expressed as;

\mathtt{216\ =2^{3} \times 3^{3}}



(b) Arrange the factors in pair of three.

\mathtt{216\ =2^{3} \times 3^{3}}

In the above prime factorization, factors 2 & 3 are arranged in pair of three since both the factors have same power 3.

This tells that the number 216 is a perfect cube.



(c) Now remove the power and multiply the existing factors.

Cube root of 216 ⟹ 2 x 3

Cube root of 216 ⟹ 6

Hence, 6 is the cube root of number 216.

This means that if you multiply 6 by itself thrice, you will get 216.

Example 02
Find the cube root of 1331

Solution
(i) Do the prime factorization

cube root solved problem

The above factorization can be expressed as;
\mathtt{1331\ =11^{3}}



(ii) Arrange the factor with power of three

\mathtt{1331\ =11^{3}}

In the above factorization, we get factor 11 with power 3. Hence, the number 1331 is a perfect cube.


(iii) Remove the power and multiply the factors

cube root of 1331 = 11

Hence, cube of 1331 is 11.

Example 3
Find the cube root of number 8000

Solution
Follow the below steps;

(i) Do the prime factorization

how to find cube root using prime factorization method


The prime factorization is expressed as;

\mathtt{8000\ =2^{3} \times 2^{3} \times 5^{3}}



(ii) Arrange all the factors in pair of three.

In the above factorization calculation, you can observe that;

Factor 2 ⟹ form pair of three
Factor 2 ⟹form pair of three
Factor 5 ⟹ form pair of three

\mathtt{8000\ =2^{3} \times 2^{3} \times 5^{3}}

Since all factors have power 3, the number 8000 is a perfect cube.



(iii) Remove the power and multiply the factors.

Cube root of 8000 ⟹ 2 x 2 x 5

Cube root of 8000 ⟹ 20

Hence, cube root of number 8000 is 20.

Example 04
Find the cube root of 9261

Solution
(i) Calculate the prime factorization

Find cube root of perfect cube number

The above factorization can be expressed as;

\mathtt{9261\ =3^{3} \times 7^{3}}



(ii) Arrange all factors in pair of three

Note that in above factorization both the factors 3 and 7 form pair of three.

Factor 3 ⟹ form pair of three
Factor 7 ⟹form pair of three

\mathtt{9261\ =3^{3} \times 7^{3}}

Since all the factors have power of three, the number 9261 is a perfect cube.



(iii) To find cube root, remove the power and simply multiply the factors.

cube root 9261 ⟹ 3 x 7

cube root 9261 ⟹ 21

Hence, cube root of number 9261 is 21.

Example 5
Find the cube root of 128

Solution
(i) Do the prime factorization

Cube root solved problems


The above factorization can be expressed as;

\mathtt{128\ =2^{3} \times 2^{3} \times \ 2}



(ii) Arrange factors in pair of three

\mathtt{128\ =2^{3} \times 2^{3} \times \ 2}

From the above expression, note that all factors do not form pair of three. This means that the number 128 is not a perfect cube.

To find the cube root, the number has to be perfect cube. Hence, we stop the process at this point.

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