Question

I need help on understanding this one! Thank you!

Answer 1

(6^⅕) (cos(-24°) + i sin(-24°))

Explanation:

First, we convert from Cartesian to polar:

r = √((-3)² + (-3√3)²)

r = √(9 + 27)

r = 6

θ = atan( (-3√3) / (-3) ), θ in the third quadrant

θ = atan(√3)

θ = 240° + 360° k

Notice that θ can be 240°, 600°, 960°, etc.

Therefore:

-3 − 3√3 i = 6 (cos(240° + 360° k) + i sin(240° + 360° k))

Now we take the fifth root:

[ 6 (cos(240° + 360° k) + i sin(240° + 360° k)) ]^⅕

(6^⅕) [ (cos(240° + 360° k) + i sin(240° + 360° k)) ]^⅕

Applying de Moivre's Theorem:

(6^⅕) (cos(⅕ × 240° + ⅕ × 360° k) + i sin(⅕ × 240° + ⅕ × 360° k))

(6^⅕) (cos(48° + 72° k) + i sin(48° + 72° k))

If we choose k = -1:

(6^⅕) (cos(-24°) + i sin(-24°))