I am trying to prove that $2^{2^{1/2}}$ is irrational, but I am having a hard time doing that. Any hints?
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1Haven't seen a 'simple' proof of this. It follows from a general theorem (Gelfond and Schneider). See also https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant – Winther Jun 05 '17 at 04:35
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...but it turns out it exists, see A short proof that $2^{\sqrt{2}}$ is not rational from the MO question Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem – Winther Jun 05 '17 at 04:39