How to raise -1 to non-integer powers How do you calculate $(-1)^x$ where $x$ is some real number. For example, what is $(-1)^{\sqrt{5}}$. This question came as I was trying to computer $e^{i\pi a}$ where $a$ is irrational.

I used this very successfully on page 7 of http://zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf
One value is
$$ (-1)^x = \cos \pi x + i \sin \pi x   $$
That was enough for my article, as this complex number can be used in Gelfond-Schneider. The conclusion is that if $x$ is real, irrational, but algebraic, then 
$\cos \pi x + i \sin \pi x$ is transcendental. I used one possible version of a contrapositive: I had both (real) $x$ and $\cos \pi x + i \sin \pi x$  algebraic, therefore $x$ was rational.
Note, however, that there are countably infinite logarithms of $(-1),$ so there are countably infinite different values of $(-1)^x.$ That's just life.
