Find the real & imaginary parts of (-sqrt(3) + i)^200? PLEASE HELP!!!?

Question

also, how do you factorise (z^5) + 100000 into real linear and quadratic factors?

Answer 1

First, convert to polar form:

-√3 + i = 2 cis (5π/6)

Now, use DeMoivre's theorem:

(2 cis (5π/6))^200 =  2^200 * cis(5π/6 * 200)

= 2^200 cis (500π/3)

= 2^200 cis (83 * 2π + 2π/3)

= 2^200 cis (2π/3)

= 2^200 (-0.5 + i√3/2)

= -2^199 + i√3 * 2^199