Roots of Complex Numbers:
The properties of the exponential operation can be used to find the nth roots of a complex number.
Example-1: To find all sixth roots of 2, we let reiθ be an arbitrary sixth root of 2 and solve for r and θ. If
then it follows that r = 21/6 R and θ = 0 solve this equation. So the real number 21/6 · ei0 = 21/6 is a sixth root of two. This is not terribly surprising, but we are not finished. We may also solve
This gives us the number
as a sixth root of two. Similarly, we can solve
to obtain the other four sixth roots of 2:
These are in fact all the sixth roots of 2.
Example-2 Let us find all third roots of i. We begin by writing i as
Solution: Solving the equation
then yields r = 1 and θ = π/6. Next, we write i = ei5π/2 and solve