 Can an irrational number raised to an irrational power be oliviayychengwh 2022-01-06 Answered
Can an irrational number raised to an irrational power be rational?
If it can be rational, how can one prove it?

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There is a classic example here. Consider $$\displaystyle{A}=\sqrt{{2}}^{{\sqrt{{2}}}}$$. Then $$\displaystyle{A}$$ is either rational or irrational. If it is irrational, then we have $$\displaystyle{A}^{{\sqrt{{2}}}}=\sqrt{{2}}^{{{2}}}={2}$$
Not exactly what you’re looking for? Juan Spiller
Consider, for example, $$\displaystyle{2}^{{\frac{{1}}{\pi}}}={x}$$ where $$\displaystyle{x}$$ should probably be irrational but $$\displaystyle{x}^{\pi}={2}$$
More generally, $$\displaystyle{2}$$ and $$\displaystyle\pi$$ can be replaced by other rational and irrational numbers, respectively. Vasquez

For example:
$$\sqrt{2}^{2\log_23}=3$$