If \((x - \frac{1}{x})\)= √6, and x > 1, what is the value of \((x^8 - \frac{1}{x^8})\)?

Given:

x - (1/x) = √6

Formula used:

x⁸ - (1/x⁸) = {x⁴ + (1/x⁴)} × {x² + (1/x²)} × {x + (1/x)} × {x - (1/x)}

If x - (1/x) = a, then x + (1/x) = √(a2 + 4)

Calculation:

x - (1/x) = √6

x² + (1/x²) = (√6)² + 2 = 8

Square both sides:

x⁴ + (1/x⁴) = (8)² - 2 = 62

Using, If x - (1/x) = a, then x + (1/x) = √{(√a)2 + 4} .

x + (1/x) = √{(√6)² + 4} = √10

Substituting the values, we get

x8 - (1/x8) = {x4 + (1/x4)} × {x2 + (1/x2)} × {x + (1/x)} × {x - (1/x)}

⇒ 62 × 8 × √10 × √6 = 496 × 2 × √15 = 992√15

∴ The correct answer is 992√15.  

 Shortcut Trick

If a - 1/a = m,

• Then a² +1/a² = m² + 2

• Then a⁴ + 1/a⁴ = m⁴ + 4m² + 2

if a2 +1/a2 = m , then a +1/a = √(m+2) 

⇒ x - 1/x =√6

⇒ x²+1/x² = (√6)² +2 = 8

⇒ x +1/x = √10

⇒ x⁴+ 1/x⁴ = (√6)⁴+4(√6)² +2 = 36+24+2 = 62 

⇒ x8 - (1/x8) = {x4 + (1/x4)} × {x2 + (1/x2)} × {x + (1/x)} × {x - (1/x)}

⇒ 62 × 8 × √10 × √6 = 496 × 2 × √15

⇒ 992√15