What are the fourth roots of 6+6√3i

Answer 1

Let put in polar form: r = (6² + (6sqrt3)²)^1/2 = 12, tanθ = 6sqrt3/6 = sqrt3, θ = π/3.

So in polar form 6 + 6sqrt3·i = 12(cos(π/3 +2πn) + isin(π/3 + 2πn)). Then forth root gives (12)^1/4(cos(π/12 + πn/2) + isin(π/12 + πn/2)) and we give n = 0, 1, 2, 3 we get 4 values of this fourth root.