Question

should be multiplied with $a^3$ $b^2$ to make it a perfect cube, if a and b are prime numbers?

Answer 1

$a^3$ $b^2$ = a $\times$ a $\times$ a $\times$ b $\times$ b
A perfect cube is a result of multiplying a number three times by itself, or, in other words, it is an integer to the exponent 3.

$a^3$ $b^2$ can be a perfect cube if 'b' is a perfect cube in the first place. But since it is given 'a' and 'b' are prime numbers, b can not be a perfect cube.

As b has only two as exponent on itself, if we multiply $a^3$ $b^2$ with one more 'b' it becomes a perfect cube.