It is true that an irrational number to the power of an irrational number could be rational. I am curious about the following:
Are there conditions that imply irrational to the power of irrational will be rational ?
2026-04-01 12:52:35.1775047955
Irrational to the Power of Irrational
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We can find a class of examples using the fact that, for $k$ integer, $\ln k$ is irrational (as a consequence of Gelfond-Schneider theorem) and $e^{\ln k}=k$ is rational.