Logic verification: $x^3$ is irrational, then $x$ is also irrational

Question

Prove, by contraposition, if $x^3$ is irrational, then $x$ is also irrational.

Just a verification do I need to show that given $x$ is rational $x^3$ is also rational?

Suppose $x \in \mathbb{Q}$ than $x=\dfrac{p}{q}$, where $p,q \in \mathbb{Z}$ and $x^3=\dfrac{p\cdot p \cdot p}{q \cdot q \cdot q}=\dfrac{p^3}{q^3}$. Since the product of\[6pt] two integers is a integer, We conclude $x^3 \in \mathbb{Q}$

Answer 1

Suppose $x \in \mathbb{Q}$ than $x=\dfrac{p}{q}$, where $p,q \in \mathbb{Z}$ and $x^3=\dfrac{p\cdot p \cdot p}{q \cdot q \cdot q}=\dfrac{p^3}{q^3}$. Since the product of two integers is a integer, We conclude

$x^3 \in \mathbb{Q}$