Algebra: question about x^x?

Question

Suppose X is a rational number. Can you prove that X^X is irrational unless x is an integer?

Answer 1

Suppose x is a rational number. Then x = p/q, where p is an integer and q is a positive integer, and p and q are in lowest terms. Then x^x = (p/q)^(p/q) = qth root of (p^p/q^p). This is irrational, unless (p^p/q^p) is a perfect qth power. But this is impossible, since we required p and q to be in lowest terms.

There are a few steps I'm skipping here, but this is the gist of the argument. In particular, you would want to justify the last sentence.

Answer 2

yes, just choose any integer plug in