TITLE: Prove that every rational solution of $x^n=c$ is integer
QUESTION [1 upvotes]: Prove that every rational solution of $x^n=c$ is integer
$c,n\in \mathbb N$

My start:
Let $x=u/q\quad \text{ such that } \gcd(u,q)=1$
$$\left( \frac u q \right)^n=c $$
I realy don't understand what should I prove here, since $c\in \mathbb N$ so $q$ must be equale to $\pm 1$, I can't find a formal way to show that.

REPLY [0 votes]: Suppose there exists a rational solution $x$ which is not an integer such that $x^n=c$, let $x=\frac{p}{q}$, where $p\in Z$ with $p\neq 0$, and $q\in N$ with $q\geq 2$. Then we may let $\gcd(p,q)=1$ and then
$$c=\left(\frac{p}{q}\right)^n=\frac{p^n}{q^n}\quad\mbox{or}\quad
  p^n=q^nc,$$
whcih follows that $q^n$ divides $p^n$, thus $2\leq q^n=\gcd(p^n,q^n)=[\gcd(p,q)]^n=1^n=1$. This gives a contradiction.
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