Math Is Fun Forum

lovelyl

Member

Registered: 2009-12-12

Posts: 2

complex numbers

given that  z = 8i express z in the polar form and obtain the three cube roots of 8i in polar form.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: complex numbers

The polar form is

, since if you look at z = 8i, it is pointing straight up.

So we have

Let

be a cube root of z, then

(last step is an application of De Moivre's theorem)

Now equating, we have

,

So the cube roots are of the form:

,

When

,

,

,

If we choose other k values, the solutions will just repeat, since w is periodic. We choose the solutions with argument

, by convention. In total, there are 3 cube roots of z: