<p>If ω is an imaginary fifth root of unity, then find the value of \(\log_2 |1 + ω + ω^2 + ω^3 - 1/ω|\).</p>
Step-by-Step Solution
Key Concept: Use the property that ω is a fifth root of unity (ω⁵ = 1) to simplify the sum 1 + ω + ω² + ω³, then express 1/ω as ω⁴ and evaluate the magnitude geometrically or algebraically.
<p><strong>Step 1:</strong> Since ω is a fifth root of unity: ω⁵ = 1, so ω ≠ 1.</p><p><strong>Step 2:</strong> Use the geometric series formula. For fifth roots of unity: 1 + ω + ω² + ω³ + ω⁴ = 0, therefore:</p><p>1 + ω + ω² + ω³ = -ω⁴</p><p><strong>Step 3:</strong> Since ω⁵ = 1, we have 1/ω = ω⁻¹ = ω⁴</p><p><strong>Step 4:</strong> Substitute into the expression:</p><p>1 + ω + ω² + ω³ - 1/ω = -ω⁴ - ω⁴ = -2ω⁴</p><p><strong>Step 5:</strong> Find the magnitude:</p><p>|-2ω⁴| = 2|ω⁴| = 2·|ω|⁴ = 2·1⁴ = 2 (since |ω| = 1 for any root of unity)</p><p><strong>Step 6:</strong> Calculate the logarithm:</p><p>log₂|1 + ω + ω² + ω³ - 1/ω| = log₂(2) = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1