Proof of rationality using product lemma

Question

Prove that if $x$ is prime then $x^{3/2}$ is irrational. Is this the correct way to prove this, or is a proof by contradiction preferable?

Using the lemma that the product of a nonzero rational number and irrational number is irrational.

Proof: Since $x^{3/2}=x^{1/2}x$ and $x^{1/2}$ is easily proved to be irrational and $x$ is prime and rational. Therefore the product $x^{3/2}$ is rational.

Any ideas?

Answer 1

Yes. As straightforward as your proof already is, I think perhaps you can also recall how you proved $\sqrt{2}$ is irrational, ie, proof by contradiction. I think it would be more straightforward in this case.

Answer 2

Yes. I believe your proof is probably one of the most concise versions of it, and it fully exploits the main idea behind proving irrationality.