Find the fifth root of 243(cos 300° + i sin 300°).

Solution:

Let Z = 243(cos 300° + i sin 300°)

Taking fifth root on both sides

Z^1/5 = (243)^1/5 [cos 300° + i sin 300° ]^1/5

Using De- Moivers theorem

If z = r [cos θ + i sinθ] then zⁿ = rⁿ[cos nθ + i sin nθ]

z^1/5 = (3⁵)^1/5 [cos(300°/5)+ i sin (300°/5)]

z^1/5 = 3[cos 60+ i sin 60°]

Find the fifth root of 243(cos 300° + i sin 300°).

Summary:

The fifth roots of 243(cos 300° + i sin 300°) is 3 [cos 60°+ i sin 60°]