Question

Complex numbers & DeMoivre's theory If z=-1-√3 i and z^6=a+bi, a=_____ and b=_____

Answer 1

Convert $z$ to exponential/polar/trigonometric form:

$r=|z|=\sqrt{(-1)^2+(-\sqrt 3)^2}=2$

$\tan\theta=\dfrac{-\sqrt3}{-1}\implies\theta=\dfrac{4\pi}3$

$\implies z=-1-\sqrt3\,i=2e^{4\pi/3\,i}=2\left(\cos\dfrac{4\pi}3+i\sin\dfrac{4\pi}3\right)$

By DeMoivre's theorem,

$z^6=\left(2e^{4\pi/3\,i}\right)^6=2^6e^{24\pi/3\,i}=64e^{8\pi\,i}$

$z^6=64\left(\cos8\pi+i\sin8\pi)=64$

so that $a=64$ and $b=0$.