Multiply 210125 by the smallest number so that the product is a perfect cube. Also, find out the cube root of the product.

Given:

210125

To do:

We have to find the smallest number by which 210125 must be multiplied so that the product is a perfect cube and find the cube root of the product.

Solution:  

Prime factorisation of 210125 is,

$210125=5\times5\times5\times41\times41$

$=5^3\times41^2$

Grouping the factors in triplets of equal factors, we see that $41^2$ is left.

In order to make 210125 a perfect cube, we have to multiply it by $41$.

$210125\times41=5^3\times41^2\times41$

$=5^3\times41^3$

$\sqrt[3]{8615125}=\sqrt[3]{5^3\times41^3}$

$=5\times41$

$=205$

The smallest number by which 210125 must be multiplied so that the product is a perfect cube is 41 and the cube root of the product is 205.