I have two questions: 1) Say I want to find solutions to 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 ?
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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.
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} ???
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