The roots of

Question

\[z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\]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.

Answer 1

I think I can read this as

z^7 = 1/√2 - 1/√2 i

so, that means

z^7 = cis 315°

z = cis(315/7)° = cis45° + k*360/7 for k=1..6