If x + \(\frac{1}{x}\) = \(2\sqrt 5 \:\) where x > 1, then the value of x³ - \(\frac{1}{{{x^3}}}\) is:

Given :

 x + \(\frac{1}{x}\) = \(2√ 5 \:\) where x > 1,

Formula used:

(a + b)² = a² + b² + 2ab

(a - b)2 = a2 + b2 - 2ab

(a - b)³ = a³ - b³ - 3ab (a - b)

Calculation:

 (x + \(\frac{1}{x}\) )² = x² +  (\(\frac{1}{x}\) )² + 2 

20 =  x2 +  (\(\frac{1}{x}\) )2 + 2 

x2 +  (\(\frac{1}{x}\) )2 = 20 - 2 =18 .....(1)

(x - \(\frac{1}{x}\) )2 = x2 +  (\(\frac{1}{x}\) )2 - 2  , Put the value of equation (1),

 (x - \(\frac{1}{x}\) )2 = 18 - 2 = 16

 (x - \(\frac{1}{x}\) ) = 4 ....(2)

apply cube power to equation 2 both side,

 => (x - \(\frac{1}{x}\) )³ = 4³

=> (x - \(\frac{1}{x}\) )3 =  x3 - \(\frac{1}{{{x^3}}}\)  - 3( x - \(\frac{1}{x}\) ) = 64, put the value of equation (2)

=>  x3 - \(\frac{1}{{{x^3}}}\)  - 3( 4 ) = 64

=> x3 - \(\frac{1}{{{x^3}}}\)  = 64 + 12 

=> x3 - \(\frac{1}{{{x^3}}}\)  = 76

Hence, '76' is the correct answer.

Shortcut Trick

If x + \(\frac{1}{x}\) = a then x - 1/x = √(a² - 4)

If  x - 1/x = k then x³ - 1/x³ = k³ + 3k

Here, x + \(\frac{1}{x}\) = \(2\sqrt 5 \:\) then  x - 1/x = √(20 - 4) = 4, then x3 - 1/x3 = (4³ + 12) = 76