<s>[INST] Need help solving this equation with complex numbers $$(z^3 + 1)^3 = 1$$
where $z$ is an element of the complex number system.
Can someone show me the most efficient way of finding all the solutions for $z$ here and also if possible please demonstrate how both sides of the equation can be converted into polar coordinates and then the equation be solved? 
Thanks a lot!!
 [/INST] As $\displaystyle(z^3+1)^3=1=\cos0+i\sin 0,$
using this, $\displaystyle z^3+1=\cos\dfrac{(2k\pi+0)}3+i\sin\dfrac{(2k\pi+0)}3$ where $k=0,1,2$
$\displaystyle\implies z^3=\cos\dfrac{(2k\pi+0)}3+i\sin\dfrac{(2k\pi+0)}3-1$
Using Double-Angle Formulas,
$\displaystyle z^3=2i\sin\frac{k\pi}3\cos\frac{k\pi}3-2\sin^2\frac{k\pi}3$
$\displaystyle=2i\sin\frac{k\pi}3\left(\cos\frac{k\pi}3+i\sin\frac{k\pi}3\right)$
$\displaystyle=2\sin\frac{k\pi}3\left(\cos\frac\pi2+i\sin\frac\pi2\right)\left(\cos\frac{k\pi}3+i\sin\frac{k\pi}3\right)$ as $\displaystyle\cos\frac\pi2+i\sin\frac\pi2=i$
$\displaystyle\implies z^3=2\sin\frac{k\pi}3\left[\cos\left(\frac\pi2+\frac{k\pi}3\right)+i\sin\left(\frac\pi2+\frac{k\pi}3\right)\right]$
Again apply this
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