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