If x is a positive real number such that x⁸+(1/x)⁸=47, then the value of x⁹+(1/x)⁹ is:

Options

1. 40√5
2. 36√5
3. 30√5
4. 34√5

Correct Answer

34√5

Explanation

Step 1: Express in terms of x⁴ + 1/x⁴ (x⁴ + 1/x⁴)² = x⁸ + 1/x⁸ + 2 → (x⁴ + 1/x⁴)² = 47 + 2 = 49 x⁴ + 1/x⁴ = 7 Step 2: Express in terms of x² + 1/x² (x² + 1/x²)² = x⁴ + 1/x⁴ + 2 → (x² + 1/x²)² = 7 + 2 = 9 x² + 1/x² = 3 Step 3: Express in terms of x + 1/x (x + 1/x)² = x² + 1/x² + 2 → (x + 1/x)² = 3 + 2 = 5 x + 1/x = √5 Step 4: Compute x³ + 1/x³ (x + 1/x)³ = x³ + 1/x³ + 3·(x + 1/x) → (√5)³ = x³ + 1/x³ + 3·√5 x³ + 1/x³ = (√5)³ − 3·√5 = 5√5 − 3√5 = 2√5 Step 5: Compute x⁹ + 1/x⁹ (x³ + 1/x³)³ = x⁹ + 1/x⁹ + 3·(x³ + 1/x³) → (2√5)³ = x⁹ + 1/x⁹ + 3·(2√5) 8·5√? Wait, compute carefully: (2√5)³ = 8·5√5 = 40√5 3·(x³ + 1/x³) = 3·2√5 = 6√5 x⁹ + 1/x⁹ = 40√5 − 6√5 = 34√5

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