Question

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

Answer 1

Prime factorising $392$, we get,

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

$=2^3\times 7^2$.

We know, a perfect cube has multiples of $3$ as powers of prime factors.

Here, number of $2$'s is $3$ and number of $7$'s is $2$.

Therefore, $392$ is clearly not a perfect cube.

Here, we need to multiply another $7$ to the factorization to make $392$ a perfect cube.

Hence, the smallest number by which $392$ must be multiplied to obtain a perfect cube is $7$.