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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Expert Answer

Ella Williams
Answered 2022-01-07 Author has 1305 answers
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 \(\displaystyle{A}^{{\sqrt{{2}}}}=\sqrt{{2}}^{{{2}}}={2}\)
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Juan Spiller
Answered 2022-01-08 Author has 5192 answers
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.
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Vasquez
Answered 2022-01-11 Author has 8850 answers

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

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