Cube roots of 8 cis(60°) are which of the following?

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Multiple Choice

Cube roots of 8 cis(60°) are which of the following?

The key idea is that for a complex number in polar form r cis θ, its cube roots have magnitude ∛r and angles (θ + 360°k)/3 for k = 0, 1, 2. Here ∛8 = 2 and the argument is 60°, so the angles are (60° + 360°k)/3 = 20°, 140°, and 260°. The roots are 2[cos20° + i sin20°], 2[cos140° + i sin140°], and 2[cos260° + i sin260°]. These match the given answer because the angles are exactly 20°, 140°, and 260° with magnitude 2. Other options misplace the angles by not dividing the original angle correctly into thirds or by using angles that don’t follow the 20° plus successive 120° steps.

The key idea is that for a complex number in polar form r cis θ, its cube roots have magnitude ∛r and angles (θ + 360°k)/3 for k = 0, 1, 2. Here ∛8 = 2 and the argument is 60°, so the angles are (60° + 360°k)/3 = 20°, 140°, and 260°. The roots are 2[cos20° + i sin20°], 2[cos140° + i sin140°], and 2[cos260° + i sin260°].

These match the given answer because the angles are exactly 20°, 140°, and 260° with magnitude 2. Other options misplace the angles by not dividing the original angle correctly into thirds or by using angles that don’t follow the 20° plus successive 120° steps.

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