What are the three cube roots of unity (solutions to z^3 = 1) in exponential form?

Answer and explanation

Correct answer: D. z = 1, e^(i 2pi/3), and e^(i 4pi/3), evenly spaced by 120 degrees on the unit circle

  1. Step 1: In polar form: 1 = 1 e^(i 2kpi). Taking cube roots: z_k = e^(i 2kpi / 3) for k = 0, 1, 2. Step 2. This yields z_0 = 1, z_1 = e^(i 2pi/3) = -0.5 + isqrt(3)/2, and z_2 = e^(i 4pi/3) = -0.5 - isqrt(3)/2. Answer: z = 1, e^(i 2pi/3), and e^(i 4pi/3), evenly spaced by 120 degrees on the unit circle. Why not the others:* The roots form an equilateral triangle inscribed in the unit circle.

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