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We know there are two non-real imaginary numbers like $a$, $b$ such that the power $a^{b}$ is a real number. For example we have $i^{i}=\frac{1}{\sqrt{e^{\pi}}}$.

Question: Are there two non-real imaginary numbers like $a, b$ such that $a^{b}$ is a natural/rational number?

  • Put $x=0,a^2+b^2=1$ here (http://math.stackexchange.com/questions/201991/for-what-values-alpha-for-complex-z-lnz-alpha-alpha-lnz/202164#202164) – lab bhattacharjee Nov 24 '13 at 18:52

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Yes, of course, but, due to the Gelfond-Schneider theorem, they would have to be transcendental.

Lucian
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