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

The roots are $e^{2k\pi i/7}$ ($k=0,\ldots,6$). Write them as $r^k$ where $r = e^{2\pi i/7}$. So the sum is (using the simple formula for summing a finite geometric series)
$$\sum_{k=0}^6 r^k = \frac{1-r^7}{1-r} = \frac{1 - e^{2\pi i}}{1-e^{2\pi i/7}} = 0$$.
