# Complex roots and de Moivre's formula

1. Sep 11, 2011

### Jonmundsson

1. The problem statement, all variables and given/known data
Find all the roots of $z^{6} = -2$

2. Relevant equations
$z = \sqrt[n]{r} (cos(\frac{\theta}{n} + k\cdot \frac{2\pi}{n}) + isin(\frac{\theta}{n} + k\cdot \frac{2\pi}{n})$ where $k = 0, 1, \cdots, n - 1$

3. The attempt at a solution
I know that $r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2} = 2$ but I don't understand how I find $\theta$. $arg z = arg -2$ isn't a nice number to work with and now I'm stumped. This is the only thing holding me back since the formula is pretty straightforward so if any one could tell me how I find $\theta$ that would be great.

2. Sep 11, 2011

### Hootenanny

Staff Emeritus
Welcome to Physics Forums.

Perhaps it would be easier if you wrote the equation in exponential notation:

$$r^6 e^{i6\theta} = 2 e^{i\pi}$$

Can you take it from here?

3. Sep 11, 2011

### Jonmundsson

Wow! Of course! Thanks for the quick help!

4. Sep 11, 2011

### Hootenanny

Staff Emeritus
A pleasure