It is given that a is a positive integer and the nth root of a is rational. A rational number can be expressed in the form p/q where p and q are integers.
a^(1/n) = p/q
=> a = (p/q)^n
If p is not an integral multiple of q, p^n is not an integral multiple of q^n, in which case p^n/q^n cannot be an integer. But we have (p/q)^n = a which is an integer.
Therefore p/q has to be an integer.
This proves that the nth root of a is an integer.