Use De Moivre’s Theorem to simplify each expression. Write the answer in the form a + bi.

Question

{1 - i(/3)}^4

Answer 1

I will assume you know the theorem and the common variables used in it

r = √(1^2 + (1/3)^2 )
= √(1 + 1/9)
= √(10/9)
√10/3

let the angle be Ø
cosØ = 1/(√10/3) = 3/√10
sinØ = (-1/3) / (√10/3) = -3/√10 , so Ø is in IV and Ø = 341.565°

so (1 - (1/3) i )^4 = (√10/3)^4 [ cos 4(341.565° + i sin 4(341.564°)
= (100/81) [ .28 + (-.96i) ]
= 28/81 - 96/81i
= 28/81 - 32/27 i
=