Question

If x-1/x = 4 then find the value of x^4 - 1/x^4

Answer 1

(x-1/x)^2 = x^2 + 1/x^2 - 2 = 16

=> x^2 + 1/x^2 = 16 + 2 = 18 .............(1)

(x + 1/x)^2 = x^2 + 1/x^2 + 2 = 20 [ From (1)]

(x + 1/x) = √20 or -√20

Now the given expression

x^4 - 1/x^4

= (x^2 + 1/x^2) ( x^2- 1/x^2)

= (x^2 + 1/x^2) ( x + 1/x) (x - 1/x)

= 18 ×( +-√20) × (4)

= 144√5 or -144√5

Answer 2

$x - \frac{1}{x} = 4...(1) \\ ( {x - \frac{1}{x} })^{2} = 16 \\ ( {x + \frac{1}{x} })^{2} - 4 = 16 \\ ( {x + \frac{1}{x} })^{2} = 20 \\ x + \frac{1}{x} = 2 \sqrt{5}...(2) \\ ( {x + \frac{1}{x} })^{2} = 20 \\ {x}^{2} + \frac{1}{ {x}^{2} } = 18...(3) \\ \\ {x}^{4} - \frac{1}{ {x}^{4} } = ( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{2} - \frac{1}{x} ) \\ = 18(x - \frac{1}{x} )(x + \frac{1}{x} ) \\ = 18 \times 4 \times 2\sqrt{5} \\ = 144 \sqrt{5}$