Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?

Question

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to be irrational or rational.

Could someone enlighten me? Thanks!

Could you provide a source (link) for the proof that you mentioned? It sounds interesting. — 727, Sep 13 '15 at 02:29
This is the square root of the Gelfond-Schneider constant, which is known to be transcendental. — Lucian, Sep 13 '15 at 09:25
Related posts: Deciding whether $2^{\sqrt2}$ is irrational/transcendenta and irrationality of $\sqrt{2}^{\sqrt{2}}$. — Martin Sleziak, Sep 16 '15 at 07:50

score 5 · Accepted Answer · answered Sep 13 '15 at 02:30

5

It's transcendental by the Gelfond-Schneider theorem.

answered Sep 13 '15 at 02:30

Qiaochu Yuan

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Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?

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