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## Main Question or Discussion Point

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?

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?