<p>If <em>α</em> and <em>β</em> are imaginary cube roots of unity, then find the value of <em>α</em><sup>4</sup> + <em>β</em><sup>4</sup> + 1/(<em>αβ</em>).</p>
Step-by-Step Solution
Key Concept: The imaginary cube roots of unity satisfy ω³ = 1 and 1 + ω + ω² = 0, where the relationship between α and β determines their product and powers efficiently.
<p><strong>Step 1:</strong> Let ω = e^(2πi/3) and ω² = e^(4πi/3) be the imaginary cube roots of unity. Then α = ω and β = ω² (or vice versa).</p><p><strong>Step 2:</strong> Key properties: ω³ = 1, ω² + ω + 1 = 0, and αβ = ω · ω² = ω³ = 1.</p><p><strong>Step 3:</strong> Compute α⁴ = ω⁴ = ω³ · ω = 1 · ω = ω = α.</p><p><strong>Step 4:</strong> Compute β⁴ = (ω²)⁴ = ω⁸ = ω⁶ · ω² = (ω³)² · ω² = 1 · ω² = ω² = β.</p><p><strong>Step 5:</strong> Therefore: α⁴ + β⁴ + 1/(αβ) = ω + ω² + 1/1 = ω + ω² + 1.</p><p><strong>Step 6:</strong> From the fundamental property 1 + ω + ω² = 0, we get ω + ω² + 1 = 0.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0