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Math Help - -1 to an exponent

  1. #1
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    -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 ?
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  2. #2
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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.
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  3. #3
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    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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