The example I am stuck with is $(-1)^.456$. According to my calculator, the answer is -1. According to WolframAlpha and the Google Calculator, it is a complex number.
WolframAlpha -- http://www.wolframalpha.com/input/?i=%28-1%29%5E.456
The way I see it, you first rewrite .456 as a fraction, which is $57/125$ in simplest form. If you were to evaluate this it would be $\sqrt[125]{(-1)^{57}}$. Based on this logic, since both the exponent and radical are odd, the result should be -1. However, more confusion comes in when I try to use .456 as a fraction not in simplest form (e.g. $114/250$). I get $\sqrt[250]{(-1)^{114}}$ and since both the radical and exponent are even, the result should be 1. Moreover, if I evaluate it alternatively like $(\sqrt[250]{-1})^{114}$, then I get a complex number.
Can anyone shed any light on this issue?
The reason that $\sqrt[250]{(-1)^{114}}$ seems to only be $1$ is that you're forgetting something: $\sqrt[n]{1}$ is always really $\pm1$, meaning that $-1$ is also a valid solution.
The reason that $(\sqrt[250]{-1})^{114}$ gives you a complex number is that the calculator is following the order of operations and finding the $250$-root of $-1$ (which is $i$) before taking that result to the $114$ power - although that should lead you to $-1$, not some multiple of $i$.
Arthur's advice, though, is the best anyone can give you: