TITLE: Can an irrational number raised to an irrational power be rational?
QUESTION [126 upvotes]: Can an irrational number raised to an irrational power be rational?
If it can be rational, how can one prove it?

REPLY [3 votes]: Let me expand orangeskid's answer, both because I think it teaches us something useful, and it might be the easiest elementary proof for this question.
The proof that $ \sqrt{2} $ is irrational is well-known, so I will not repeat it here.
But there's a proof just as simple showing that $ \log 3 / \log 2 $ is irrational.  Suppose on contrary that $ \log 3 / \log 2 = p / q $ where p and q are integers.  Since $ 0 < \log 3 / \log 2 $, we can choose $ p $ and $ q $ both as positive integers.  The equality then rearranges to $ 3^q = 2^p $.  But here, the left hand side is odd and the right hand side is even, so we get a contradiction.
That gives a positive answer to the original question:
$$
\big(\sqrt{2}\big)^{2 \log 3 / \log 2} = 3
$$
Thanks to Rand al'Thor, who mentioned this problem in SE chat and thus inspired this answer.