Evaluate (5+5√3i)^7 using DeMoivre’s theorem.

Question

Write your answer in rectangular form. Show all your work.

Answer 1

let z = 5+5√3i = 5(1 + √3i)
magnitude = 5√(1+3) = 10
tanØ = 5/√3/5 = √3
Ø = 60°

z = 10(cos60° + i sin60°)
z^7 = 10^7(cos420° + i sin 420°)
= 10^7(cos60 + isin 60) , since cos 420 = cos(360+60) = cos60 , same for sin420

= 10^7/2 + i√3 10^7/2