<p>The value of \((1 + \omega - \omega^2)^7\) is:</p>
Step-by-Step Solution
Key Concept: Recognize that ω is a cube root of unity (1 + ω + ω² = 0), which allows you to simplify 1 + ω - ω² by substituting ω² = -1 - ω, then use De Moivre's theorem on the resulting expression in polar form.
<p><strong>Step 1:</strong> Use the property of cube roots of unity: 1 + ω + ω² = 0, so 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 + 2ω = 2(1 + ω)</p><p><strong>Step 3:</strong> Since ω² = -1 - ω, we have 1 + ω = -ω². Therefore: 2(1 + ω) = 2(-ω²) = -2ω²</p><p><strong>Step 4:</strong> Calculate (-2ω²)⁷ = (-2)⁷(ω²)⁷ = -128·ω¹⁴</p><p><strong>Step 5:</strong> Since ω³ = 1, reduce the exponent: ω¹⁴ = ω¹²·ω² = (ω³)⁴·ω² = 1⁴·ω² = ω²</p><p><strong>Step 6:</strong> Therefore: -128ω² = -128(-1 - ω) = 128 + 128ω</p><p>∴ Answer: D</p>
Correct Answer: D