# Principal root of a Complex Number

1. Sep 10, 2009

### DEMJ

1. The problem statement, all variables and given/known data
Find all the roots in rectangular coordinates, exhibit them as vertices of certain squares, and point out which is the principal root.

3. The attempt at a solution

The problem is $$(-8 -8\sqrt{3}i)^{\frac{1}{4}}$$ and I found the four roots easily to be

$$\pm(\sqrt{3} - i), \pm(1 + \sqrt{3}i)$$. So when it says to exhibit them as vertices of certain squares does this mean I should just graph the four points? and it should be a square.

Also as for which point is the principal root I have no clue. I am guessing it has something to do with principal argument but I am having trouble finding much information on principal roots.

2. Sep 10, 2009

### Elucidus

For any non-zero complex number, its fourth roots are arranged on the vertices of a square. You are being asked to verify this for the given number.

If I'm not mistaken, the principal root is the root whose standardized argument is least.

--Elucidus