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.

Decoding the LaTeX that didn't render, we seek sum of the angles of the seventh roots of

That's on the unit circle, 45 degrees into the third quadrant, aka 225 degrees.

The seventh roots will all be separated by 360/7, around 51 degrees. The first seventh root has

That's around 32 degrees.

The next angle is

The next one is

and in general

The first sum is just -135° since it's one seventh of the sum of seven -135s.

We have 1+2+3+4+5+6+7 = (1+7)+(2+6)+(3+5) + 4 = 28 so

If I didn't screw it up, that means the answer is

1305°

ohh it's too hard to explain