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))

Solution:

We will be using DeMoivre's theorem to find the answer.

According to DeMoivre's theorem if z = r(cosθ + isinθ), then zⁿ = rⁿ(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)]

Hence, a fifth root of the given complex number 32(cos(π/3) + i sin(π/3)) is 2(cos(π/15) + i sin(π/15))

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

Summary:

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))