Question

x^3 -2 -2i sqrt(3) =0

Answer 1

\[\begin{align*}
x^3-2-2i\sqrt3&=0\\
x^3&=2+2i\sqrt3
\end{align*}\]
Let's convert to polar so we can use DeMoivre's theorem:
\[r^2=2^2+(2\sqrt3)^2=4+12=16~~\implies~~r=4\]
\[\begin{cases}4\cos\theta=2\\4\sin\theta=2\sqrt3\end{cases}~~\implies~~\theta=\frac{\pi}{3}\]
So you have
\[\begin{align*}x^3&=2+2i\sqrt3\\
x^3&=4\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)\\
x&=\sqrt[3]{4\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)}
\end{align*}\]
Find all the third roots of the complex number.