Using Direct Proofs in Discrete Math

Question

Use a direct proof to show that if $x$ is a rational number, then $x^2$ is also a rational number.

I know I need to use the definition of rational numbers but don't know how to do it in this problem.

Answer 1

We have $x=\frac{p}{q}$ with $p,q \in \mathbb Z$ and $q \ne 0$, hence

$$x^2=\frac{p^2}{q^2}.$$

Conclusion ?