# Complex exponential to a power

Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##.
However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for example? Then ##e^{2 \pi i (\frac{1}{2})} = e^{\pi i} = -1##. What am I doing wrong here?

SACHIN KUMAR

jedishrfu
Mentor
Isn't that Euler's identity you now have?

Add 1 to both sides to get ##e^{i\pi} + 1 = -1 + 1## hence ##e^{i\pi} + 1 = 0 ##

So sqrt of 1 has two roots 1 and -1 right? squaring the answer you got brings you back to the original ##e^{2\pi i} = 1##

@fresh_42 or @Mark44 can provide a better answer I think.

fresh_42
Mentor
SACHIN KUMAR, Mr Davis 97 and jedishrfu
jedishrfu
Mentor

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