Help With a proof (Irrational Number)

Question

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational.

Its contrapositive will be:

If $r^\frac{1}{5}$ is not irrational, then $r$ is not irrational.

How can I prove the contrapositive ?

Answer 1

If $r^{1/5}$ is rational, then there exist $p,q \in \mathbb{Z}$ with $q\neq 0$ so that $r^{1/5}=\frac{p}{q}$. Therefore $r = \frac{p^5}{q^5}$ is a rational number since $\mathbb{Q}$ is closed under product.

Answer 2

Suppose $r^{1/5}$ rational. Then $r^{1/5}=\frac{p}{q}$ for certain $p,q\in\mathbb Q$. We can suppose $\gcd(p,q)=1$. Then $r=(r^{1/5})^5=\frac{p^5}{q^5}\in\mathbb Q$

Q.E.D.