Can we find a number, let's say $x$ that's not a simple fraction, and if you raise it to the power of $π$ or raise $π$ to the power of $x$, both results end up being not simple fractions too?
2026-03-29 09:11:34.1774775494
Does there exist an irrational number $x$ such that both $x^π$ and $π^x$ are also irrational?
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Following Akiva's advice, I'm posting this as a full answer, but I'll leave the details to you:
If for any irrational, atleast one of $x^\pi$ or $π^x$ is rational, then the set of irrationals injects in a union of two countable sets: Which ones ? Why is this absurd ?