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Evaluate the value of [(-1 + √-3)/2]¹⁰⁰ + [(-1 - √-3)/2]¹⁰⁰.

1. 2
2. 0
3. -1
4. 1

Correct answer: -1

Solution

(-1 +/- sqrt(-3))/2 are the complex cube roots of unity omega and omega^2. Since 100 ≡ 1 and 200 ≡ 2 (mod 3), the sum is omega^100 + (omega^2)^100 = omega + omega^2 = -1.

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