Given that $z=1+i$, find the value of $n\in\mathbb{Z^+}$ such that $z^n$ is real.

Question

Given that $z=1+i$, find the smallest value of $n\in\mathbb{Z^+}$ such that $z^n$ is real.

I'm wondering if there's an algebraic way of solving this question, aside from the obvious trial and error method.

Using the trial and error method:
$$(1+i)^2=2i$$
$$(1+i)^3=-2+2i$$
$$(1+i)^4=-4$$

Hence, $n=4$.

Answer 1

Hint:

Use the polar form of $z$ and apply de Moivre's theorem