<p>Let <em>a</em> be a real number satisfying \(a^3 + \frac{1}{a^3} = 18\). Then find the value of \(a^4 + \frac{1}{a^4}\).</p>
Step-by-Step Solution
Key Concept: Use the identity chain: from a³ + 1/a³, first find a + 1/a, then derive a² + 1/a² and a⁴ + 1/a⁴ using the algebraic relation (x + y)² = x² + y² + 2xy.
<p><strong>Step 1:</strong> Let y = a + 1/a. We need to find y first.</p><p>We know: (a + 1/a)³ = a³ + 1/a³ + 3(a + 1/a)</p><p>So: y³ = 18 + 3y, which gives y³ - 3y - 18 = 0</p><p><strong>Step 2:</strong> Factor: (y - 3)(y² + 3y + 6) = 0</p><p>Since y² + 3y + 6 has discriminant 9 - 24 = -15 < 0, the only real solution is y = 3.</p><p>Therefore: a + 1/a = 3</p><p><strong>Step 3:</strong> Find a² + 1/a². Using (a + 1/a)² = a² + 1/a² + 2:</p><p>9 = a² + 1/a² + 2</p><p>So: a² + 1/a² = 7</p><p><strong>Step 4:</strong> Find a⁴ + 1/a⁴. Using (a² + 1/a²)² = a⁴ + 1/a⁴ + 2:</p><p>49 = a⁴ + 1/a⁴ + 2</p><p>∴ a⁴ + 1/a⁴ = <strong>47</strong></p>
Correct Answer: 47