Which of the following is a fifth root of the given complex number 32 (cos(π/3) + i sin(π/3))?

2(cos(7π/15) + i sin(7π/15))

2(cos(π/15) + i sin(π/15))

2(cos(11π/15) + i sin(11π/15))

32(cos(π/15) + i sin(π/15))
We will be using DeMoivre's theorem to find the answer.
>Answer: A fifth root of the given complex number 32(cos(π/3) + i sin(π/3)) is (b) 2(cos(π/15) + i sin(π/15))
Let us solve it step by step.
Explanation:
According to DeMoivre's theorem if z = r(cosθ + isinθ), then z^{n} = r^{n}(cos nθ + i sin nθ)
It has also been proved that it is true for all n∈Q.
z = 32(cos(π/3) + i sin(π/3))
z^{1/5} = 32^{1/5} [cos(π/3 × 1/5) + i sin(π/3 × 1/5)]
z^{1/5} = 2[cos(π/15) + i sin(π/15)]