Question

If x = 1 - √2, find the value of `( x - 1/x )^3`

Answer 1

Given that x = 1 - √2
We need to find the value of `( x - 1/x )^3`

Since x = 1 - √2, we have 
`1/x = 1/( 1 - sqrt2) xx ( 1 + sqrt2 )/( 1 + sqrt2 )`

⇒ `1/x = (1 + sqrt2)/( (1)^2 - (sqrt2)^2 )`      [ Since ( a - b )( a + b ) = a² - b² ]

⇒ `1/x = [ 1 + sqrt2 ]/[ 1 - 2 ]`

⇒ `1/x = [ 1 + sqrt2 ]/-1`

⇒ `1/x = -( 1 + sqrt2 )`              .....(1)

Thus, `( x - 1/x ) = ( 1 - √2 ) - (-( 1 + sqrt2))`

⇒ `( x - 1/x ) = 1 - √2 + 1 + √2`

⇒ `( x - 1/x ) = 2`

⇒ `( x - 1/x )^3 = 2^3`

⇒ `( x - 1/x )^3 = 8`