Under what conditions can the exponent product rule $a^{nm}=(a^n)^m$ be used. Under what conditions can you use the exponent product rule $a^{nm}=(a^n)^{^m}$?
For example $i=\sqrt{-1}=(-1)^{1/2}=(-1)^{2\times 1/4}=((-1)^2)^{1/4}=(1)^{1/4}=1$ 
What's going wrong here?

What's wrong in your string of equalities is that:


*

*$i=\sqrt{-1}$ is wrong since $i$ is one of the square roots of $-1$ (the other one being $-i$, of course);

*$(-1)^{2\times(1/4)}=\bigl((-1)^2\bigr)^{1/4}$ is wrong because $(-1)^{2\times(1/4)}$ can be $i$ or $-i$, whereas $\bigl((-1)^2\bigr)^{1/4}$ can be one of four different numbers ($\pm1$ and $\pm i$);

*$1^{1/4}=1$ is wrong because $1^{1/4}$ can be any fourth root of $1$: again, $\pm1$ and $\pm i$.

