---
tag: 0XWCKJZRYH
kind: statement
role: theorem
name: A root of an integer is an integer or irrational
msc: [11J72]
uses: [080T2TXVQV, 0C4JS9RF5M]
sources:
  - url: https://en.wikipedia.org/wiki/Irrational_number
    locator: General roots
---

## Statement

Let $x$ be a real number, let $n$ be a positive integer and let $m$ be an [[integer|0C4JS9RF5M]] with $x^n = m$.
If $x$ is not an integer, then $x$ is [[irrational|080T2TXVQV]].

## 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 $5/2$.