Complex constant from single root.

[DmytriE]

Homework Statement

Do not use a calculator for this problem. Express your answers using square roots and/or fractional multiples of x.

Determine the complex constant c such that v is a root of: z⁶ - c = 0

Homework Equations

v = $\sqrt{3} - j$

The Attempt at a Solution

I believe the following are true:
1. There are 3 distinct roots for this equation.
2. Each of the distinct roots will have a conjugate pair to ensure there is no middle term.
3. If 2 is true, then two of the six roots are $\sqrt{3} -j$ and $\sqrt{3} +j$

z⁶ - c = 0 --> z⁶ = c. Yes, this is as far as I have gotten. I'm not sure how to figure out the other 4 roots.

 

[rude man]

c = z⁶
v = √3 - j
Use re^jθ = r(cosθ + jsinθ)
Write v as exponential
Solve for c