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
Best Answer
Well, by DeMoivre's Theorem, we have $$(\operatorname{cis}\theta_k)^7=\operatorname{cis}(7\theta_k)$$ for $k=1,...,7.$ We need for $$(\operatorname{cis}\theta_k)^7=\operatorname{cis} 225^\circ,$$ as you've already determined, and so we need $$\operatorname{cis}(7\theta_k)=\operatorname{cis} 225^\circ\\\frac{\operatorname{cis}(7\theta_k)}{\operatorname{cis} 225^\circ}=1\\\operatorname{cis}(7\theta_k-225^\circ)=1$$ for $k=1,...,7.$ Can you take it from there?