Question

Can an irrational power of an irrational
number be rational? (complex numbers should be omitted)

Answer 1

yes

Answer 2

proof
consider
\[\sqrt{2}^{\sqrt{2}}\]
if it is rational, then you know of one
if it is irrational then consider
\[(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=\sqrt{2}^2=2\]which is rational
in either case there must be one