How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational?

Question

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function...

I tried my hand at the proof by contradiction. Assuming $\sqrt{2}^{\sqrt{2}}$ is rational. Hence,

$\sqrt{2}^{\sqrt{2}} = \frac{p}{q} \implies \big({\frac{p}{q}}\big)^{\sqrt{2}} = 2$

Can I now say, that we know that $\sqrt{2}$ is irrational, but $\frac{p}{q}$ and $2$ are rational;

If $a$ is rational and $b$ is irrational, $a^b$ must be irrational. *(1)

In our case $a^b = 2$ which is rational: a contradiction to our original assumption.

This is just tomfoolery on my part. I don't know if *(1) is even true or not; how can I approach such a problem?

Answer 1

The irrationality of $\sqrt{2}^{\sqrt{2}}$ is a trivial consequence of the Gelfond-Schneider theorem.