Find the fifth roots of 1024 Write your answers in polar for

Find the fifth roots of -1024. Write your answers in polar form, then convert to rectangular form.

Solution

Because this is a fifth root there should be 5 solutions, all equally spaced on a circle in the complex plane. We know that 2 is one of the solutions because -4^5 = 5, so the circle has a radius of 4

To find the other solutions, first determine the angle of each vector. They must divide the circle into 5 pieces so they must be (1 circle divided into 5 pieces) 360/5 = 72 degrees apart.

They list out as: 72, 144, 216, 288, 360

The 360 brings us back to the real line, and this is our original solution (2).

The others are given as:

radius (given by real solution) --->4 * [cos(x) + i sin(x)] <----- direction in complex plane

= 4 cos(x) + 4 i sin(x)