Question

Use Demoivres Theorem to find (-square root 3 +i)^6

Answer 1

Answer:

$z=(-\sqrt{3}+i)^6$ = -64

Step-by-step explanation:

You have the following complex number:

$z=(-\sqrt{3}+i)^6$       (1)

The Demoivres theorem stables the following:

$z^n=r^n(cos(n\theta)+i sin(n\theta))$      (2)

In this case you have n=6

In order to use the theorem you first find r and θ, as follow:

$r=\sqrt{3+1}=2\\\\\theta=tan^{-1}(\frac{1}{\sqrt{3}})=30\°$

Next, you replace these values into the equation (2) with n=6:

$z^6=(2)^6[cos(6*30\°)+isin(6*30\°)]\\\\z^6=64[-1+i0]=-64$

Then, the solution is -64