Suppose $x^a = x^b$ is integral for two integers $a,b$ and $\operatorname{gcd}(a,b) = 1$. Is $x$ an integer? This is the final step in my proof for a larger homework problem - if I can prove that this statement is true, then I have finished my other proof - however my mind has gone blank and I cannot solve this. Any pointers?
Suppose $x^a, x^b$ are integral for two positive integers $a,b$ and $\operatorname{gcd}(a,b) = 1$. Is $x$ an integer? 

By Bezout's Theorem, there exist integers $s$ and $t$ such that $as+bt=1$. It follows that $x^1=(x^a)^s(x^b)^t$. Thus $x$ is rational.
But if $x$ is rational and not an integer, and $n$ is a positive integer, then $x^n$ cannot be an integer. This follows from the Rational Roots Theorem. For if $c$ is an integer, the only rational solutions of $x^n-c=0$ are of the form $k/l$, where $k$ divides $c$ and $l$ divides $1$. In particular, we must have $l=\pm 1$. 
