1. ## complex numbers

Hello, would really appreciate if anyone could answer this questions for me ... i'm really stuck! If possible, could you show all working out because otherwise i will get lost!

* Find all the fifth roots of - 1.

* Find all the third roots of 2 + 2 i .

*3. If w = (z - i)/(z+i) and z lies below the real axis, show that w lies outside the unit circle
| w | = 1.
How will w move as z travels along the real axis from - infinity to + infinity ?

*4. Prove that the area A ( a, b, c ) of the triangle in the complex plane with corners at a, b, c
must be C (Complex), ordered in anti-clockwise fashion, is given by the formula:
A ( a, b, c ) = (i/4)( ab - ab + b c - bc + c a - ca )

Thank you !

2. You can't get ANY of these?

1 and 2 require ONLY DeMoivre. Convert the values to polar coordinates and you should see it.

The third looks like an algebra problem. It could be a little messy, I guess.

Let's see if you can get through those while I think about #4, or someone else chimes in.

3. the first two questions were dealt with here. i like Soroban's method the most

4. Note that $(-1)^{1/5} = -1$. If you let $\zeta = \cos \frac{2\pi}{5}+i\sin \frac{2\pi }{5}$. Then the roots are $-1,-\zeta,-\zeta^2.-\zeta^3.-\zeta^4$.