Prove if $\sqrt r$ is rational, then $r$ is also rational

Question

How would I prove that if $\sqrt r$ is rational, then $r$ is also rational?

I started assuming that $\sqrt r$ is rational then $\sqrt r=\frac{p}{q}$ where $\gcd(p,q) = 1$. Then $r$ would be $p^2/q^2$ but ... how can I prove that is rational?.

Answer 1

Solution:

Suppose $\sqrt{r}$ is rational. By definition, then, can find some $p,q \in \mathbb{Z}$ such that $\sqrt{r} = \frac{p}{q} $. This means that

$$ r = \frac{ p^2}{q^2} = \frac{ p \cdot p}{q \cdot q} \in \mathbb{Q} $$

and thus the result follows.