If \omega\neq1 is a cube root of unity, then what is (1+\omega-\omega^{2})^{100}+(1-\omega+\omega^{2})^{100} equal to?

  1. A. 2^{100}\omega^{2}
  2. B. 2^{100}\omega
  3. C. 2^{100}
  4. D. -2^{100} ✓

Correct Answer: D. -2^{100}

Explanation

Using 1+\omega+\omega^2 = 0, substitute 1+\omega = -\omega^2 and 1+\omega^2 = -\omega. The expression becomes (-2\omega^2)^{100} + (-2\omega)^{100} = 2^{100}(\omega^{200} + \omega^{100}). Since \omega^3 = 1, we get 2^{100}(\omega^2 + \omega) = 2^{100}(-1) = -2^{100}.

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