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Question
If `[x^2 + 1]/x = 3 1/3 "and x > 1; Find If" x^3 - 1/x^3`
Solution
Given `[x^2 + 1]/x = 3 1/3`
`[x^2 + 1]/x = 10/3`
`[x + 1/x] = 10/3`
Squaring on both sides, we get
`x^2 + 1/x^2 + 2 = 100/9`
`x^2 + 1/x^2 = [ 100 - 18 ]/9 = 82/9`
`x - 1/x = sqrt[( x + 1/x )^2 - 4] = sqrt( 100/9 - 4 ) = sqrt( 64/9) = 8/3`
∴ `x - 1/x = 8/3`
Cubing both sides, we get
`( x - 1/x )^3 = 512/27`
`x^3 - 1/x^3 - 3( x - 1/x ) = 512/27`
`x^3 - 1/x^3 = 512/27 + 8 = [ 512 + 216 ]/27 = 728/27`
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