Question

Solve using demoivres theorem

Answer 1

It's difficult to make out, but I think the task is to expand


`(\sqrt3-i)^4`

Write the number in polar form first:


`\sqrt3-i=2e^(-i\pi/6)`

By DeMoivre's theorem, you have


`(\sqrt3-i)^4=\left(2e^(-i\pi/6)\right)^4=2^4e^(-i4\pi/6)=16e^(-i2\pi/3)`

and converting back to Cartesian form, this number is equivalent to


`16\left(\cos\frac{2\pi}3-i\sin\frac{2\pi}3\right)=16\left(-\frac12+-\frac{\sqrt3}2\right)=-8(1+i\sqrt3)`