Express the roots of unity in standard for a+bi. Determine z and the four fourth roots of z if one fourth root of z is -2 - 2i.
According to DeMoivre Theorem, the 4th root of a number
is given by `z_k=root(4)(r)(cos[(theta+2kPi)/4]+isin[(theta+2kPi)/4])`
Looking at the -2-2i=2(-1-i) this means that the sin and cos are equal, hence theta is either Pi/4 or Pi/4+Pi (1st and 3rd quadrant sin and cos have the same sign)
Since we both sin and cos are negative in the given problem then theta has to equal 5Pi.
The other roots can be obtained by adding 2Pi, 4Pi, and 6Pi to 5Pi. I will work one for you.
(To help you see geometrically, 7Pi/4 falls in the 4th quadrant, 9Pi/4=2Pi+Pi/4 falls in the 1st, and 11Pi/4=2Pi+3Pi/4 fall in the 2nd quadrant)