Find all solutions of the equation Show all step Leave answe

Find all solutions of the equation. Show all step: Leave answers in exact trigonometric form.

Find all solutions of the equation. Show all step: Leave answers in exact trigonometric form. x^4 - 16 i = 0

Solution

x^4 = 16i
x^4 = 0 + 16i
x^4 = 16 * (0 + i)
x^4 = 2^4 * (cos(pi/2 + 2pi * k) + i * sin(pi/2 + 2pi * k))
x =2 *( (cos((pi/2) * (1 + 4k)) + i * sin((pi/2) * (1 + 4k)))^(1/4)
x = 2 * (cos((pi/8) * (1 + 4k)) + i * sin((pi/8) * (1 + 4k))

k is an integer

x =>
2 * cos(pi/8) + 2i * sin(pi/8)
2 * cos(3pi/8) + 2i * sin(3pi/8)
2 * cos(5pi/8) + 2i * sin(5pi/8)
2 * cos(7pi/8) + 2i * sin(7pi/8)