Use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi

Question

(1−√3i​)^5

Answer 1

There are 5 fifth roots of 1 - √3 i

First, convert to polar form: r = √[(1)² + (√3)²] = 2.

Since the point (1, -√3) lies in quadrant 4, θ = 360° - Tan^-1(√3) = 300°

Polar form of 1 - √3i is 2(cos300° + isin300°)

By DeMoivre's Theorem, the five 5th roots of 1 - √3i are equally spaced about the circle centered at (0,0) with radius 2^1/5

First fifth root: 2^1/5(cos(300°/5) + isin(300°/5)) = 2^1/5(√3 + 0.5i)

360°/5 = 72°

Second fifth root: 2^1/5(cos(60°+72°) + isin(60°+72°)) = 2^1/5(cos132° + isin132°)

Third fifth root: 2^1/5(cos(132° + 72°) + isin(132° + 72°)) = 2^1/5(cos204° + isin204°)

Fourth fifth root: 2^1/5(cos(204° + 72°) + isin(204° + 72°) = 2^1/5(cos276° + isin276°)

Fifth fifth root: 2^1/5(cos(276° + 72°) + isin(276° + 72°)) = 2^1/5(cos348° + isin348°)

Answer 2

z = 1 - i√3

||z|| = 2

θ = 60º

z⁵ = 2⁵(cos 300º + i sin 300º)