Subtle Proof Error I'm having trouble seeing the error in the following "proof":
$$ (-4)=(-4)^1=(-4)^\frac{2}{2}=[(-4)^2]^\frac{1}{2}=[16]^\frac{1}{2}=4$$ therefore $(-4)=4$. 
Obviously this is incorrect, but I'm not seeing where the error is occuring. I appreciate any help. Thanks.

The rule $a^{bc}=(a^b)^c$ is not true for arbitrary $a$, $b$, and $c$. The most important cases where it works are


*

*When $b$ and $c$ are both integers.

*When $a$ is a nonnegative real, and $b$ and $c$ are reals.


Here you're applying it with $a=-4$ and $c=\frac12$, which does not fall into any of those cases. Therefore you don't have any guarantee that the rule holds in this case, and indeed it doesn't.
