Find all the complex roots Leave your answers in polar form

Find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of -4i. z_k = [cos(degree + degree k)+i sin (degree + degree k)], k = 0, 1, ..., (Type an exact answer in the first answer box. Type any angles in degrees between 0 degree and 360 degree.

Solution

z = ( -4i)^1/4

z = r e^theta = 4e^i*3pi/2

Use DeMoivre\'s Theorem :

So, z = 4(cos3pi/2 + isin3pi/2)

z^1/4 = 4^1/4[cos(3pi/2 +2kpi)/4 +isin(3pi/2 +2kpi)/4 ]

z_k= 4^1/4[ cos(3pi/8 +k*pi/2) +i*sin(3pi/8 +k*pi/2)