1. ## De moivres theorem.

Use DMT to solve the following equations in polar form.

$\displaystyle z^3=-1-i$
$\displaystyle z^3=-4+4\sqrt{3i}$

2. What have you tried so far?. These problems are just routine knowing the corresponding theory.

3. I'll show you how to do one, then you try to do the other...

$\displaystyle \displaystyle |-1-i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$.

Note that this complex number is in the third quadrant. The angle in the first quadrant is $\displaystyle \displaystyle \arctan{\frac{1}{1}} = \frac{\pi}{4}$.
So the angle in the third quadrant, $\displaystyle \displaystyle \arg{(-1 - i)} = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$.

$\displaystyle \displaystyle z^3 = \sqrt{2}\,\textrm{cis}\,{\left(\frac{5\pi}{4}\righ t)}$

$\displaystyle \displaystyle z = (\sqrt{2})^{\frac{1}{3}}\,\textrm{cis}\,{\left(\fr ac{1}{3}\cdot \frac{5\pi}{4}\right)}$

$\displaystyle \displaystyle = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12}\right)}$.

There are always three cube roots, and they are of the same magnitude and evenly spaced about a circle. So they differ by an angle of $\displaystyle \displaystyle \frac{2\pi}{3}$.

So the other solutions are $\displaystyle \displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12}\right)}$

and $\displaystyle \displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{21\pi}{12}\right)}$.

But since $\displaystyle \displaystyle -\pi < \arg{z} \leq \pi$, we rewrite this as

$\displaystyle \displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(-\frac{11\pi}{12}\right)}$ and $\displaystyle \displaystyle \sqrt[6]{2}\,{\left(-\frac{3\pi}{12}\right)}$.