Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gelfond–Schneider theorem.

We know that ${\sqrt2}^{\sqrt2}$ is a transcendental number by the Gel'fond-Schneider's theorem. I've tried to prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem, but I'm facing difficulty. I need your help.

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