Question

Is 64 a perfect cube?

Answer 1

Hint: To check whether the given number is a perfect cube or not, we will find out its cube root. If its cube root will be an integer, then the number will be a perfect cube and if the cube root of that number will be a fraction or decimal, then the number will not be a perfect square.

Complete step-by-step answer:
In this question, we are given a number and we need to determine if it is a perfect cube or not.
A perfect cube is a number obtained by multiplying a same number three times. Let us say we take a number x, then its perfect cube will be ${x^3}$. To find out whether the given number is a perfect cube or not, we find the cube root of that given number. If we get the cube root as an integer, then it is a perfect cube and if we get the cube root in fraction or decimal, than that number would not be a perfect cube.
For example: 8. If we find the cube root of 8, we get 2 which is a natural number. Hence, 8 is a perfect cube.
So, to determine whether 64 is a perfect cube or not, we need to find its cube root.
Steps to find the cube root of a number:
Step 1:
Prime factorize 64.
$ \Rightarrow 64 = 2 \times 32$
           $
   = 2 \times 2 \times 16 \\
   = 2 \times 2 \times 2 \times 8 \\
   = 2 \times 2 \times 2 \times 2 \times 4 \\
   = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\
 $
Step 2:
$
   \Rightarrow 64 = {2^6} \\
   \Rightarrow 64 = {\left( {{2^2}} \right)^3} \\
   \Rightarrow 64 = {\left( 4 \right)^3} \\
   \Rightarrow \sqrt[3]{{64}} = \sqrt[3]{{{4^3}}} = 4 \\
 $
Hence, the cube root of 64 is 4. Hence, it is an integer. So, we can say that 64 is a perfect cube.

Note: Here, the $\sqrt {} $ sign is known as the radical and the value written inside this sign is known as the radicand.
Note that we can find cube and cube roots of negative integers too.
For example: Cube of -5 will be $ - 5 \times - 5 \times - 5 = - 125$.
And cube root of $ - 8$ will be $ - 2$.