Question

Is 392 a perfect cube? If not, find the smallest natural number by which 392 should be multiplied so that the product is a perfect cube.

Answer 1

The prime factorization of 392 gives:

$392=2\times 2\times 2\times 7\times 7$

Since, we can see, number 7 cannot be paired in a group of three. Therefore, 392 is not a perfect cube.

To make it a perfect cube, we have to multiply the 7 by the original number.

Thus,

$2\times 2\times 2\times 7\times 7\times 7=2744,$

which is a perfect cube, such as $2^3\times 7^3$ or ${14}^3$

Hence, the smallest natural number which should be multiplied to 392 to make a perfect cube is 7.