If $x^3 =7$, then $x$ is irrational

Question

Assuming there is a real number $x$ with $ x^3 =7$, prove that $x$ is irrational.

I started the proof by contradiction, and I got to the point that $7q^3 = p^3$, but I don't know what should I do after this.

Answer 1

Hint: You can assume that $p, q$ are coprime. From $7q^3 = p^3$, you have $ p \mid 7q^3$. What can you deduce from this?

This proof will be very similar to the standard proof that $\sqrt 2$ is irrational.