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.
To find the roots of $z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$, we can express the right-hand side in polar form. Let's call $z_0 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$, and write it in polar form as $z_0 = r_0\operatorname{cis}\theta_0$, where $r_0$ is the magnitude and $\theta_0$ is the argument of $z_0$.
To find $r_0$, we compute the magnitude as $r_0 = |z_0| = \sqrt{\left(-\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1$.
To find $\theta_0$, we compute the argument as $\theta_0 = \operatorname{arg}(z_0) = \operatorname{arg}\left(-\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right)$. Since $z_0$ lies in the third quadrant, its argument is $\theta_0 = \operatorname{arg}(z_0) = \operatorname{arg}\left(-\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right) = \arctan\left(\frac{-\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}}\right) = \arctan(1) = 45^\circ$.
Now that we have the polar form of $z_0$, we can write it as $z_0 = \operatorname{cis} 45^\circ$.
The roots of the equation $z^7 = z_0$ will have a magnitude of $r = \sqrt[7]{r_0}$ and argument values $\theta = \frac{\theta_0 + 360^\circ k}{7}$, where $k = 0, 1, 2, \dots, 6$.
Plugging in the values, we get:
$\theta_1 = \frac{\theta_0 + 360^\circ \cdot 0}{7} = \frac{45^\circ}{7}$
$\theta_2 = \frac{\theta_0 + 360^\circ \cdot 1}{7} = \frac{45^\circ + 360^\circ}{7}$
$\theta_3 = \frac{\theta_0 + 360^\circ \cdot 2}{7} = \frac{45^\circ + 2 \cdot 360^\circ}{7}$
$\vdots$
$\theta_7 = \frac{\theta_0 + 360^\circ \cdot 6}{7} = \frac{45^\circ + 6 \cdot 360^\circ}{7}$
To find the sum of all the $\theta_k$, we can use the formula for the sum of an arithmetic series:
$\theta_1 + \theta_2 + \dots + \theta_7 = \frac{7}{2}\left(\theta_1 + \theta_7\right) = \frac{7}{2}\left(\frac{\theta_0 + 360^\circ \cdot 0}{7} + \frac{\theta_0 + 360^\circ \cdot 6}{7}\right)$
Simplifying, we get:
$\theta_1 + \theta_2 + \dots + \theta_7 = \frac{7}{2}\left(\frac{2\theta_0 + 360^\circ \cdot 6}{7}\right) = \frac{7}{2}\left(\frac{2 \cdot 45^\circ + 360^\circ \cdot 6}{7}\right)$
$\theta_1 + \theta_2 + \dots + \theta_7 = 45^\circ + 360^\circ \cdot 6 = \boxed{2340^\circ}$