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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?

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up vote 1 down vote accepted

You've been a little careless in writing down $z$. I'd write it as $z=18\sqrt2e^{5\pi i/4}$. Then the 7th roots are $\root7\of{18\sqrt2}\zeta e^{5\pi i/28}$ where $\zeta$ runs through the 7th roots of unity. Now you should be able to figure out which $\zeta$ to take.

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