Question

find the natural number by which 392 must be multiplied so that the product is a perfect cube

Answer 1

392=2×2×2×7×7

The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect cube.

Now, to make its a cube, we need one more 7. In that case

392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube.

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