Prove the irrationality of $2^{\frac{1}{n}}$ for $n > 2$
So, we suppose $2^{\frac{1}{n}}$ is rational (= $\frac{p}{q}$). Therefore, $2 = \frac{p^n}{q^n} \Rightarrow q^n + q^n = p^n$ and this contradicts Fermat's last theorem. Is this a correct proof?
What do you think about this proof?