PLEASE HELP WITH MY TRIG HW: De Moivre's Theorem *Best Answer Awarded*?

De Moivre's theorem states that if z = r(cos(x) + isin(x)) then z^n = r^n(cos(nx) + isin(nx))

So first we write z = -3√3 + 3i in the form r(cos(x) + isin(x))

r = √((-3√3)^2 + 3^2) = √(27 + 9) = √36 = 6

sin(x) = 3/6 = 1/2 so x = 2πn + π/6 or x = 2πn + 5π/6

cos(x) is negative, so π/2 < x < 3π/2

so x = 5π/6

so z = 6(cos(5π/6) + isin(5π/6))

so z^5 = 6^5(cos(25π/6) + isin(25π/6))

z^5 = 7776(cos(π/6) + isin(π/6))

z^5 = 7776(√3/2 + i/2)

z^5 = 3888√3 + 3888i

Let

z=[-3sqr(3)+3i]^5

=>

z=(3^5)[-sqr(3)+i]^5----(1)

Let z=243(z1)^5, where

z1=-sqr(3)+i=>

|z1|=sqr[3+1]=2=>

z1=2[-sqr(3)/2+i/2]=>

z1=2[cos(5pi/6)+isin(5pi/6)]

=>

(z1)^5=32[cos(25pi/6)+isin(25pi/6]

(De Moivre's Theorem)

=>

(z1)^5=32[cos(pi/6)+isin(pi/6)]

=>

(z1)^5=32[sqr(3)/2+i/2]

=>

(z1)^5=16[sqr(3)+i]

=>

z=243(16)[sqr(3)+i]

(from (1))

=>

z=3888[sqr(3)+i]