# Math Help - -1 to an exponent

1. ## -1 to an exponent

I have two questions:

1) Say I want to find solutions to $(-1)^{m/n}$ for integers m and n. When does a real solution exist to this? And when do only complex solutions exist?

2) Say x is an irrational number. What is $(-1)^x$?

2. 1) Real solutions will exist if n is odd. Else imaginary.
See it's simple. (-1)^(m/n) is merely {(-1)^(1/n)}^m
It's the mth power of the nth root of -1.

3. 2) I'll try using Demoivre's theorem here.

-1= e^{i( pi)} where i is the square root of -1 and pi is-well, pi :-P

so (-1)^x = e^{i(pi)x} ???