If (x + \(\rm \frac{1}{x}\)) = 3\(\sqrt2\), and x > 1, what is the value of (x⁸ - \(\frac{1}{x^8}\))?

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 20 Jul 2023 Shift 4)

Answer 1

Formula used

(a + b) = a² + b² + 2ab.

a⁸ - b⁸ = (a⁴ + b⁴)(a⁴ - b⁴)

(a⁴ - b⁴) = (a² + b²)(a² - b²)

Calculation

⇒ x + 1/x = 3√2

⇒ (x +1/x)² = 18

⇒ x²+ 1/x² + 2 = 18

⇒ x²+ 1/x² = 16

⇒ (x²+ 1/x²)² = 16²

⇒ x⁴+ 1/x⁴ + 2 = 256

⇒ x⁴+ 1/x⁴ = 254

⇒ x4 + 1/x4 - 2 = 254 - 2

⇒ (x2 - 1/x2)2 = 252

⇒ x²- 1/x² = 6√7

⇒ x8 - 1/x⁸ = (x4 + 1/x4)(x4 - 1/x4)

⇒ x8 - 1/x8 = (x4 + 1/x4)(x² - 1/x²)(x² + 1/x²)

= 254 × 16 × 6√7

= 24384√7

The answer is 24384√7.