Complex Numbers

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

This hard difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of complex numbers. It appeared in the 2025 exam.

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Complex Numbers

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