if $a^n$ is irrational then $a$ is irrational

Theorem.

If $a$ be a real number and $n$ is an integer such that $a^n$ is irrational, then $a$ is irrational.

Proof.

We show this by way of contrapositive. In other words, we show that, if $a$ is rational, then $a^n$ is rational.

Let $a$ be rational. Then there exist integers $b$ and $c$ with $c\ne 0$ such that $a={\displaystyle \frac{b}{c}}$. Thus, $a^n={\displaystyle \frac{b^n}{c^n}}$, which is a rational number. ∎

Note that the converse is not true. For example, $\sqrt{2}$ is irrational and ${\left(\sqrt{2}\right)}^2=2$ is rational.

Title	if $a^n$ is irrational then $a$ is irrational
Canonical name	IfAnIsIrrationalThenaIsIrrational
Date of creation	2013-03-22 14:18:50
Last modified on	2013-03-22 14:18:50
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 11J82
Classification	msc 11J72