A root of an integer is an integer or irrational

Statement

Let $x$ be a real number, let $n$ be a positive integer and let $m$ be an integer with $x^n = m$. If $x$ is not an integer, then $x$ is irrational.

Remarks

Thus, a rational number $x$ with $x^n \in \mathbb{Z}$ is an integer. For example, $\sqrt[3]{2}$ is irrational, because $1^3 < 2 < 2^3$ shows that it is not an integer. The statement follows from the rational root theorem for the polynomial $X^n - m$.

Explanations

L1 human

Take a root of a whole number, for example $\sqrt{10}$ or $\sqrt[3]{20}$. Either the root is a whole number, as for $\sqrt{9} = 3$, or it is not a fraction at all. A root of a whole number is never a fraction like $5/2$.