If \alpha=\frac{-1+\sqrt{-3}}{2} then what is the value of (1+\alpha^{19}-\alpha^{35})^{100}-(1-3\alpha^{25}+\alpha^{38})^{50}?

  1. A. -2
  2. B. -1
  3. C. 0 ✓
  4. D. 2

Correct Answer: C. 0

Explanation

Here \alpha = \omega, the complex cube root of unity. Using \omega^3 = 1 and 1 + \omega + \omega^2 = 0: \alpha^{19} = \omega, \alpha^{35} = \omega^2, \alpha^{25} = \omega, \alpha^{38} = \omega^2. First term: (1 + \omega - \omega^2)^{100} = (-2\omega^2)^{100} = 2^{100}\omega^2. Second term: (1 - 3\omega + \omega^2)^{50} = (-4\omega)^{50} = 2^{100}\omega^2. The difference is 2^{100}\omega^2 - 2^{100}\omega^2 = 0.

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