Question

Find the indicated power using De Moivre’s Theorem (−5−5sqrt 3i)^3

Answer 1

-5 - 5√3 i = -5 (1 + √3 i )

We have modulus

|-5 (1 + √3 i )| = 5 √(1² + (√3)²) = 5√4 = 10

and argument

arg(-5 - 5√3 i ) = π - arctan(√3) = 2π/3

(we subtract from π because the given complex number lies in the third quadrant of the complex plane, whereas the arctan function only returns angles between -π/2 and π/2)

so that the polar form of the number is

-5 - 5√3 i = 10 exp(2π/3 i )

By DeMoivre's theorem, we have

(-5 - 5√3 i )³ = 10³ exp(3 × 2π/3 i ) = 1000 exp(2πi ) = 1000