How To Find an Indicated Power with DeMoivre's Theorem?

Question

Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi.)

Answer 1

(-2-2i)^3 =

by simpler methods

(-2)^3 times (1+i)^3

= -8(1+3i+3i^2 + i^3)

=-8(1-3+3i-i) =

-8(-2+2i) =

16-16i

So, if you try DeMoivre's Method, correctly,

you should get the same answer

(-2-2i)^3 =

-8(1+i)^3=

-8z(cosx+isinx)^3 where sinx=cosx =1 and x = 45 degrees or pi/4

cos45=sqr2/2 which makes z = 2sqr2

-8(2sqr2)[cos45 + isin45)^3 (move the exponent 3 to a coefficient of the angle 45

-16sq2(cos3(45)+ isin3(45)) =

-16sqr2(cos135 + isin135) =

-16sqr2(-sqr2/2 + isqr2/2) =

16 - 16i

Answer 2

z = -√2 - i√2

|| z || = 2

θ = 5π / 4

z² = 2³(cos 15π / 4 + i sin 15π / 4)