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|=√((-1)^2+(-\sqrt 3)^2)=2`

`\tan\theta=(-\sqrt3)/(-1)\implies\theta=\frac{4\pi}3`

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