What is the 4th root of -8i by De morives theorem?

Answer 1

Find all solutions of z4 = -8i.

Recall:

z = a + bi, the complex form

z = |z|(cos θ + i sin θ), the polar coordinate form

So we can write:

z4 = [|z|(cos θ + i sin θ)]4 = |z|4(cos 4θ + i sin 4θ); -8i = 8(0 - i), and we have

|z|4(cos 4θ + i sin 4θ) = 8(0 - i).

Thus, |z|4 = 8, so |z|= 81/4.

The angle θ for z must satisfy cos 4θ = 0 and sin 4θ = -1.

Consequently, 4θ = 3pi/2 + 2npi for an integer n, so that θ = 3pi/8 + npi/2.

The different values of θ obtained where 0 ≤ θ ≤ 2pi are:

n = 0, θ = 3pi/8 (1st quadrant)

n = 1, θ = 3pi/8 + pi/2 = 7pi/8 (2nd quadrant)

n = 2, θ = 3pi/8 + pi = 11pi/8 ( 3rd quadrant)

n = 3, θ = 3pi/8 + 3pi/2 = 15pi/8 (4th quadrant))

Thus the solutions of z4 = -8i are

(81/4)(cos 3pi/8 + i sin 3pi8) ≈ 0.6435942529 + 1.553773974 i

(81/4)(cos 7pi/8 + i sin 7pi/8) ≈ -1.553773974 + 0.6435942529 i

(81/4)(cos 11pi/8 + i sin 11pi/8) ≈ -0.6435942529 - 1.553773974 i

(81/4)(cos 15pi/8 + i sin 15pi/8) ≈ 1.553773974 - 0.6435942529 i