<s>[INST] Complex Numbers and exponential form and roots The roots of $z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$ are $\text{cis } \theta_1, \text{cis } \theta_2, \dots, \text{cis } \theta_7,$ where $ 0^\circ \le \theta_k < 360^\circ $for all $ 1 \le k \le 7$. Find $\theta_1 + \theta_2 + \dots + \theta_7$. Give your answer in degrees.
In exponential form this is $z^7 = e^ \left(5 \pi i/4 \right)$. How should I simplify? Thanks
 [/INST] $\def\cis{\operatorname{cis}}$Let's consider a more general problem. We have $a=r\cis\alpha$ and we write its $n$th roots as
\begin{gather}
\sqrt[n]{r}\cis\frac{\beta}{n}\\
\sqrt[n]{r}\cis\left(\frac{\beta}{n}+\frac{2\pi}{n}\right)\\
\sqrt[n]{r}\cis\left(\frac{\beta}{n}+2\frac{2\pi}{n}\right)\\
\dots\\
\sqrt[n]{r}\cis\left(\frac{\beta}{n}+(n-1)\frac{2\pi}{n}\right)
\end{gather}
Can you sum up all those angles? Use $360$ instead of $2\pi$ if you so prefer or are forced to.
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