What are the differences between $r^n\text{cis}(\theta)$ and $r^n \text{cis}(\frac{2k\mathrm\pi}n)$?

Question

I have some questions to describe the solutions in polar form and complex form. And, some answers are calculated by $r^n \text{cis}\left(\frac{2k\mathrm\pi}n\right)$ and some are calculated by $r^n \text{cis}(\theta)$. And, some are calculated by $r^n \text{cis}(n\ \theta)$

Can you tell me which formula are used when?

Answer 1

All these depend on $\theta$.

For example, for integer $n$ and non-negative $r$: $$(r(\cos\theta+i\sin\theta))^n=r^n(\cos{n\theta}+i\sin{n\theta}),$$ which for $\theta=\frac{2\pi k}{n}$, where $n\neq0$ is integer and $k$ is integer gives $$(r(\cos\theta+i\sin\theta))^n=r^n.$$