Convert [tex]z[/tex] to exponential/polar/trigonometric form:
[tex]r=|z|=\sqrt{(-1)^2+(-\sqrt 3)^2}=2[/tex]
[tex]\tan\theta=\dfrac{-\sqrt3}{-1}\implies\theta=\dfrac{4\pi}3[/tex]
[tex]\implies z=-1-\sqrt3\,i=2e^{4\pi/3\,i}=2\left(\cos\dfrac{4\pi}3+i\sin\dfrac{4\pi}3\right)[/tex]
By DeMoivre's theorem,
[tex]z^6=\left(2e^{4\pi/3\,i}\right)^6=2^6e^{24\pi/3\,i}=64e^{8\pi\,i}[/tex]
[tex]z^6=64\left(\cos8\pi+i\sin8\pi)=64[/tex]
so that [tex]a=64[/tex] and [tex]b=0[/tex].
