Question

If `(x^2 + 1)/x = 3 1/3` and  x > 1; find If `x^3 - 1/x^3`

If `(x^2 + 1)/x = 3 1/3  "find"  x^3 - 1/x^3`

Answer 1

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 + 1/x)^2 = (10/3)^2`

`x^2 + 1/x^2 + 2 = 100/9`

`x^2 + 1/x^2 = (100 - 18)/9`

`∴x^2 + 1/x^2 = 82/9`

Also,

`x - 1/x = sqrt((x+1/x)^2 - 4)`

= `sqrt(100/9 - 4)`

= `sqrt(64/9)`

∴ `x - 1/x = 8/3`

Cubing both sides, we get

`(x - 1/x)^3 = (8/3)^3`

`x^3 - 1/x^3 - 3(x - 1/x) = 512/27`

`x^3-1/x^3-3(8/3)=512/27`

`x^3 - 1/x^3 - 8 = 512/27`

`x^3 - 1/x^3 = 512/27 + 8`

`x^3 - 1/x^3 = (512 + 216)/27`

∴ `x^3 - 1/x^3 = 728/27`