TITLE: Does $\exp(2ir\pi)$ equal $1$? What's wrong?
QUESTION [8 upvotes]: Since $e^{ix}=\cos x+i\sin x$,
thus  $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$   
Now I take arbitrary real number $r$
then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$  
But this cannot be true since $\cos2\pi r+i\sin2\pi r$  isn't necessarily $1$ for arbitrary $r$. Where did it go wrong?

REPLY [6 votes]: The answer is simple:
$$(a^b)^c = a^{bc}$$
is an equality which only holds for real numbers with $a>0$ and does not necesarily hold on complex numbers.