Can the exponentiation of an integer by a rational be a non-integer rational?

Question

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$.

My question is the following: what is the nature of $n^r$?

My first guess is that $n^r$ is an integer or an irrational but cannot be a non-integer rational. Is it correct? And if not, can you provide an example where $n^r$ is a non-integer rational?

Answer 1

Your intuition is correct: $n^r$ is a root of $x^q-n^p=0$. By the rational root theorem, $n^r$ is either an integer or irrational.

Answer 2

Write $n^{p/q} = (n^p)^{1/q}$ and call $n^p = m$. You are asking about the nature of $x = m^{1/q}$, well it's a number that satisfies $x^q -m = 0$. Now take a look at the Rational root theorem.