TITLE: Unexpected result from Euler's formula
QUESTION [10 upvotes]: I am a bit confused with a result I get from Euler's formula:
$e^{2\pi i} = 1$
$\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$
$(e^{2\pi i})^{\frac{1}{3}}= 1$
$e^{\frac{2}{3} \pi i} = 1$
This last result seems problematic, since the left-hand side of the previous equation should result in...:
$e^{\frac{2}{3} \pi i} = −\frac{1}{2} + \frac{\sqrt{3}}{2}i$
... given that $e^{x i} = \cos(x) + i\sin(x)$
So is there a mistake in my first manipulations? If not, how can one interpret the results?

REPLY [5 votes]: You are making the classical mistake of generalizing the exponentiation rule $(a^b)^c=a^{bc}$ to complex numbers: it doesn't hold !