# 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 $c\neq 0$ such that $\displaystyle a=\frac{b}{c}$. Thus, $\displaystyle a^{n}=\frac{b^{n}}{c^{n}}$, 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 IfAnIsIrrationalThenaIsIrrational 2013-03-22 14:18:50 2013-03-22 14:18:50 Wkbj79 (1863) Wkbj79 (1863) 13 Wkbj79 (1863) Theorem msc 11J82 msc 11J72