Prove that if $a, b, n\in \mathbb{N}, n\geq2\longrightarrow \sqrt[\leftroot{-2}\uproot{2}n]{a}\in \mathbb{Q} \iff a=b^n$.

Question

Prove that if $a, b, n\in \mathbb{N}, n\geq2\longrightarrow \sqrt[\leftroot{-2}\uproot{2}n]{a}\in \mathbb{Q} \iff a=b^n$.

I'm at a complete loss here, I tried using the order of a prime function but didn't get anywhere. Any tips to get going?

Thanks in advance!

Answer 1

This is an immediate consequence of the Rational Root Test. Apply it to the polynomial $\,x^n\!-a.$