In questions like this that give and ask information about divisibility of higher powers of an integer, it's a good idea to
start with the prime factorized form of the integer itself. That is, as the first step, express
\(P =\)\(P1^{a}*P2^{b}*P3^{c}*P4^{d}\) . . . where P1, P2, P3, P4 etc. are prime numbers and a, b, c, d etc. are non-negative integers.
This means
\(P^2\) =\(P1^{2a}*P2^{2b}*P3^{2c}*P4^{2d}\) . . .
We are given that \(P^2\) is divisible by \(12=\) \(2^{2}*3^{1}\)
This means, P1 = 2 and 2a >= 2
That is, a > = 1. So, minimum possible value of a = 1
Also, P2 = 3 and 2b > = 1
That is, b > = 0.5
But b must be an integer. So, minimum possible value of b = 1
So, we see that P = \((2^{1}*3^{1})(something)\) . . .
We don't have any idea about what the value of this 'something' is, but we can say for sure that P must be divisible by \((2^{1}*3^{1})\) and so, \(P^3\) must be divisible by \((2^{3}*3^{3})\)
Hope this was useful!
Japinder