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So this is my thought process:

$$ \begin{align} (e^{i(\frac{2\pi}{3})})^n &= (e^{i(\frac{2\pi n}{3})}) \\ &= \cos{(\frac{2\pi n}{3})} + i \sin{(\frac{2\pi n}{3})} \\ \end{align} $$

This is real when

$$ \begin{align} \sin{(\frac{2\pi n}{3})} &= 0 \\ \frac{2\pi n}{3} &= k\pi, \; k \in \mathbb Z \\ n &= \frac{3k}{2}, \; k \in \mathbb Z \\ \end{align} $$

But my books says that the answer is

$$n = 3k, \; k \in \mathbb Z^+$$

Which I don't understand. What am I doing wrong? (Or is the book perhaps wrong?)

Skeleton Bow
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    In the problem statement, is $n$ required to be an integer? – Sloan Oct 10 '16 at 21:16
  • Then you should restrict your $n$ to be in $\mathbb{Z}^{+}$, in which case you and the book agree. – Sloan Oct 10 '16 at 21:21
  • your k must be even. – hamam_Abdallah Oct 10 '16 at 21:22
  • @AbdallahHammam It does seem so! However, the question is asking for the values for which it is real, not necessarily those for which it is real and positive. So wouldn't my method be correct? – Skeleton Bow Oct 10 '16 at 21:24
  • @Sloan I just checked again, and it doesn't seem like it necessarily restricts $n$ to be positive (this was just meant for another part of the question): but for the course I'm learning, it is safe to assume that $n$ is an integer. – Skeleton Bow Oct 10 '16 at 21:26

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