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I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$

Is there something wrong in this logic? When I plug the different versions into Mathematica, I get complex number for the first two versions, and real numbers for the last two. And, the last two are opposite in sign. What am I missing?

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up vote 7 down vote accepted

$x^y$ is multi-valued unless either $y$ is an integer or $x$ is real and positive.

In the general case one has to define $x^y = e^{y\log x}$, which inherits the ambiguity of the complex logarithm.

Even for $x>0$, $y\not\in\mathbb Z$, this general definition in principle leads to multiple complex values, but then it makes sense to adopt a convention that real logarithm of $x$ is used. In other cases there is no such common convention -- it isn't possible to fix one convention such that the exponentiation rules always hold anyway.

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