$\cos(2\pi t)=1$ for all real $t$?

Question

Can you spot my mistake in the following line? $\cos(2\pi t)=\Re(\exp(2\pi it))=\Re((\exp(2\pi i))^t)=\Re(1^t)=1$

Answer 1

You are assuming that$$e^{tz}=(e^z)^t\tag1$$here. If $z\in\mathbb R$, there is no problem here, but otherwise what does $(1)$ mean? The number $e^z$ can be any complex number (other than $0$).