Question

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

(2-2i)^7

Answer 1

(2-2i)
=2(1-i)
=2(sqrt 2) (1/sqrt 2 -1/sqrt2)
=2(sqrt 2) (cos -pi/4 + isin -pi/4)
By DeMoivre's thm,
(2-2i)^7 = 2(sqrt 2) (cos -pi/4 + isin -pi/4)
(2-2i)= 2(sqrt 2) (cos -pi/4 + isin -pi/4)^(1/7)
= 2(sqrt2)^(1/7) (cos (2pi-pi/4)/7 + i sin ( 2pi -pi/4)/7)
= 2(sqrt2)^(1/7) (cos pi/4 + isin pi/4)
=2(sqrt2)^(1/7) (1/sqrt 2 +i/sqrt 2)
= (2(sqrt2)^(1/7)) (1/sqrt 2)(1 +i)
=2(sqrt2)^(-6/7) (1 +i)

Answer 2

(2-2i) = 2(sqrt 2) (cos -pi/4 + isin -pi/4)
(2-2i)^7 = (2(sqrt 2)^7 (cos -pi/4 + isin -pi/4)^7
= (2(sqrt 2)^7 (cos -7pi/4 + isin -7pi/4)