Prove that if 'a' is a positive integer and the nth root of 'a' is rational, then the nth root of 'a' must be an integer.

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.