Question

is 392 a perfect cube? if not what is the what is the smallest natural number by which 392 must be multiplicated so that the product is a perfect cube

Answer 1

Prime factorising 392, we get,
392=2×2×2×7×7
=2
3
×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.