This came out on today's test: "Under what condition(s) is $a^b$ a real number, if $a$ is a complex number and $b$ is a positive integer?" And of course, the bonus extension: "In general, under what condition(s) is $a^b$ a real number, if both $a$ and $b$ are complex numbers?"
For the first part, I did this: $a^b = r^be^{ib\theta}$. So $b\theta = n\pi$ for some positive integer $n$, or for all integer $n$ in general, outside the context of the question. (we could visualize this using the Argand Diagram).
I have, however, no idea where to start for the second part.
It was worth 10 additional marks in the 50 marks paper though. Not sure if the teachers are really serious about it, or just trolling us.
Best Answer
Note that for $a\ne 0$ $$a^b := \exp(b \cdot \ln(a)) = \exp(b \cdot (\ln(|a|) + i\arg a)) \\ = \exp(\Re b \ln(|a|) - \Im b \arg a + i (\Re b \arg a + \Im b \ln(|a|)))$$ And $\exp(z) \in \mathbb R \Leftrightarrow \Im(z) \in \pi\mathbb Z$. so $$a^b \in \mathbb R \Leftrightarrow \Re b \arg a + \Im b \ln(|a|) \in \pi\mathbb Z$$ If $a=0$ and $\Re b > 0$, $a^b:=0$ can be defined (for $b\in\mathbb R^+$ it is "natural", if $\Im b\ne 0$, some do NOT define it).
For $a=0, \Re b\le 0$, $a^b$ is not defined.