Precalculus: Write each expression in x+iy form and re^(i theta) form

Question

Write each expression in x+iy form and re^(i theta) form.

a) [sqrt(3) (coś pi/18 +isin pi/18)]^6

b) (1-sqrt(5)i)^8

Answer 1

a) [sqrt(3) (coś pi/18 +isin pi/18)]^6 = 27(cosπ/3 + isinπ/3) = 27/2 + 27√3/2i = 27e^iπ/3

b) (1-sqrt(5)i)^8 = {√6(cos5.1329 + isin5..1329)]⁸ = 6⁴(cos41.0634 + isin41.0634) = -1264 - 286.2167i =

1296e^41.0634i

r = √(1 + 5) = √6; tanθ = -√5/1 = - √5; θ = 2π - tan^-1√5 = 5.1329