Find the cube roots of -27.
Let z = -27 = -27(cos 0∘ + isin 0∘)
Cube root z = [−27(cos0∘+isin0∘)]13
= - 3 [cos2kπ3+isin2kπ3] where k = 0,1,2
When k = 0
z = -3[cos 0 + isin 0 ] = -3
When k = 1
z = - 3 [cos2π3+isin2π3] = -3 [−12+i(√32)]
= 32 - i3√32
When k = 2
z = - 3 [cos4π3+isin4π3]
= - 3 [−12−i(√32)]
= 32 + i3√32
Cube roots are -3, 32 - 3√3i2 , 32 + 3√3i2