<p>If \(\omega\) is an imaginary cube root of unity, then \((1 + \omega - \omega^2)^7\) equals</p>
Step-by-Step Solution
Key Concept: Use the fundamental property that ω³ = 1 and 1 + ω + ω² = 0 to simplify (1 + ω - ω²) before exponentiation. Recognize that 1 + ω - ω² can be reduced to a simple form using the constraint equation.
<p><strong>Step 1:</strong> Use the constraint for cube roots of unity: 1 + ω + ω² = 0, which gives ω² = -(1 + ω)</p><p><strong>Step 2:</strong> Substitute into the expression: 1 + ω - ω² = 1 + ω - (-(1 + ω)) = 1 + ω + 1 + ω = 2(1 + ω)</p><p><strong>Step 3:</strong> Therefore (1 + ω - ω²)⁷ = [2(1 + ω)]⁷ = 2⁷(1 + ω)⁷ = 128(1 + ω)⁷</p><p><strong>Step 4:</strong> Note that 1 + ω = -ω², so (1 + ω)³ = (-ω²)³ = -ω⁶ = -(ω³)² = -1</p><p><strong>Step 5:</strong> Thus (1 + ω)⁷ = (1 + ω)⁶ · (1 + ω) = [(1 + ω)³]² · (1 + ω) = (-1)² · (1 + ω) = 1 + ω = -ω²</p><p><strong>Step 6:</strong> Final answer: 128(1 + ω)⁷ = 128(-ω²) = -128ω²</p><p>∴ Answer: D</p>
Correct Answer: D