For complex numbers $a,b,c$, explain why $a^{b\cdot c}=(a^b)^c$ is not necessarily true.

I'll provide a counterexample. Let $c=1/2$ and $b=2$. Now let $a=i$. $a^{bc}=i^{2*1/2}=i^{1}=i$. But, $a^{b}=-1$ and $(a^{b})^{c}=(-1)^{1/2}=-i \textrm{ or } i$. We see that raising a number to the one half is multivalued, so we have an issue here. So in this case, $a^{bc} \neq (a^b)^c$. The explanation for how to get around this has to do with Branch cuts (https://en.wikipedia.org/wiki/Branch_point#Branch_cuts}).