I know this may be silly but lately I've been wondering about powers of negative numbers and I came up with an ambiguous example:
$(-1)^{(\frac{22}{10})}$
I wanted to know if it was $+1$ or $-1$. My first attempt was writing it down as a root:
$\sqrt[10]{(-1)^{22}}$
Minus one to the even power is obviously plus one, then 1 to the power of $\frac{1}{10}$ is 1. Yet if we first reduce the fraction and then calculate we get a different answer:
$(-1)^{(\frac{22}{10})} = (-1)^{(\frac{11}{5})} = \sqrt[5]{(-1)^{11}}$
Even though it's (I suppose) the same situation, $(-1)^{11}$ is $-1$ and we end up with $\sqrt[5]{-1}$, a complex number. What is (or is there even) the value of $(-1)^{(\frac{22}{10})}$?