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

Question

UPSC Quizroom · Accepted Answer

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

🎯 UPSC Quiz Room