Find complex $z$ such that $z$ has the largest possible real part, and satisfies the equation:
$z^7 = -18 -18i$
So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ where $k = 0, \pm 1, \pm 2, \pm 3$
I figured the largest possible real part would be when the argument is the smallest, i.e., closest to 0. This would be when $k = 0$ and $z = (18\sqrt{2})^{\frac{1}{7}}e^{i\frac{\pi}{28}}$, but my provided answer is:
$(18\sqrt{2})^{\frac{1}{7}}e^{\frac{5\pi i}{28}}$
Have I misunderstood the question somehow? How do I get to this answer?