Question

If \(\sqrt x + \frac{1}{\sqrt x} = \sqrt 6\) then the value of \(x^6 + \frac{1}{x^6}\) will be
1. 2270
2. 2502
3. 2712
4. 2702

Answer 1

Correct Answer - Option 4 : 2702

Given:

\(√ x + \frac{1}{√ x} = √ 6\)

Formula:

(a + b)² = a² + b² + 2ab

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

\(√ x + \frac{1}{√ x} = √ 6\)

⇒ (√x + 1/√x)² = 6

⇒ x + 1/x + 2 = 6

⇒ (x + 1/x) = 4

⇒ x² + 1/x² = 4² - 2 = 14

⇒ (x² + 1/x²)³ = x⁶ + 1/x⁶ + 3(x² + 1/x²)

∴ (x⁶ + 1/x⁶) = 2702