1. ## complex number exponentials

simplify, in cartesian and polar forms:

(sqrt(3)-3i)^10

(sqrt(3)-3i)^ -1

(sqrt(3)-3i)^ 1/3

2. You must learn to find the arguments of complex numbers!
$\arg \left( {\sqrt 3 - 3i} \right) = \arctan \left( {\frac{{ - 3}}{{\sqrt 3 }}} \right) = \arctan \left( { - \sqrt 3 } \right) = - \frac{\pi }{3}$.

Now all the three questions depend upon two facts: $\arg \left( {\sqrt 3 - 3i} \right) = - \frac{\pi }{3}$ and $\left| {\sqrt 3 - 3i} \right| = \sqrt {12} = 2\sqrt{3}$

3. okay, so now what?

4. Originally Posted by mistykz
okay, so now what?
we can express the complex number $x + iy$ as $r e^{i \theta}$, where $r = |x + iy| = \sqrt {x^2 + y^2}$ and $\theta = \arg (x + iy)$

in this form, it is easy to apply the powers and simplify

you can see the "Background" section in the first post here for more information.