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Let alpha and beta be the two non-real (complex) cube roots of unity. For an integer n, find an expression for alphaⁿ + betaⁿ.

1. 2cos 2npi/3
2. cos 2npi/3
3. 2i sin 2npi/3
4. i sin 2npi/3

Correct answer: 2cos 2npi/3

Solution

Writing the roots as conjugate exponentials, their n-th powers add to twice the real part, namely 2cos(2npi/3).

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