Find the indicated roots Please show initial change to 108ra

Find the indicated roots. Please show initial change to -108rad2

For a positive integer n, the complex number z = r(cos theta + i sin theta ) has exactly n distinct nth roots given by n r(cos theta +2 pi k/n + i sin theta + 2 pi k/n) where k = 0, 1, 2, . . . , n - 1. Consider the following. Cube roots of -108 2(-1 + i) Use the formula to find the indicated roots of the complex number. (Enter your answers in trigonometric form.) k = 0: k = 1: k = 2:

Solution

Cube root of -108 rt 2(-1+i)  is = cube root of -27*8 (-1/rt2+i/rt2)

Cub root of -27 * 8 = -6

1/ rt 2(-1+i) can be written as (cos 3pi/4+isin 3pi/4)

Now take cube root and apply De Moivre theorem

= cos( 3pi/4 + 2kpi)/3 + i sin (3pi/4 + 2kpi)/3, for k = 0,1,2

Substitute k= 0,1,2

k=0 : root = cos pi/4 +isin pi/4 = (1+i)/rt 2

k =1 root =(-1+i)/rt 2

k =2 root = (-1-i)/rt 2