Complex Numbers

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If ω is a cube root of unity, then (1 + ω - ω²)⁶ equals

RankGuru · Accepted Answer

1. **Core Identity**: For cube roots of unity, $1 + \omega + \omega^2 = 0 \implies 1 + \omega = -\omega^2$.
2. **Simplify Base**: 
   - $(1 + \omega - \omega^2) = (-\omega^2 - \omega^2)$
   - $= -2\omega^2$.
3. **Exponentiation**:
   - $(-2\omega^2)^6 = (-2)^6 \cdot (\omega^2)^6$
   - $= 64 \cdot \omega^{12}$
4. **Final Result**: Since $\omega^3 = 1$, $\omega^{12} = (\omega^3)^4 = 1^4 = 1$.
   - Result $= 64 	imes 1 = 64$.

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