Question

If \(\frac{4}{3}\left(x^2 + \frac{1}{x^2}\right) = 110 \frac{2}{3}\) find \(\frac{1}{9}\left(x^3 - \frac{1}{x^3}\right)\) where x > 0.
1. 74
2. 76
3. 84
4. 85

Answer 1

Correct Answer - Option 3 : 84

Given 

\(\frac{4}{3}\left(x^2 + \frac{1}{x^2}\right) = 110 \frac{2}{3}\)

Formula Used 

If x² + 1/x² = a

then x - 1/x = √(a² - 2) = b

and also x³ - 1/x³ = b³ + 3b 

Calculation 

(4/3)(x² + 1/x²) = (332/3)

⇒ (x2 + 1/x2) = (332/3) × (3/4) 

⇒ x² + 1/x² = 83

⇒ x - 1/x = √(83 - 2) = √81 = 9 

⇒ x³ - 1/x³ = 9³ + 3(9) = 729 + 27 = 756 

⇒ 1/9(x3 - 1/x³) = 84