Roots of Complex Numbers:

The properties of the exponential operation can be used to find the n^th roots of a complex number.

Example-1: To find all sixth roots of 2, we let re^iθ be an arbitrary sixth root of 2 and solve for r and θ. If

                     

then it follows that r = 2^1/6 R and θ = 0 solve this equation. So the real number 21/6 · eⁱ⁰ = 2^1/6 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 = e^i5π/2 and solve