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

Explanation

`x = 2 + √3`
`1/x = 1/2 + √3`
`= 1 × (2 - √3)/(2 + √3) (2 - √3)`
=` (2 - √3)/(2^2 - √3^2)`
= `(2 - √3)/4 - 3`
= `(2 - √3)`
Therefore ,
`x^2 = (2 + √3)`
=` (2)^2 + (√3)^2 + 2 × 2 × √3`
=` 4 + 3 + 4√3`
=` 7 + 4√3`
`1/x^2 = (2 - √3)^2`
= `(2)^2 + (√3)^2 - 2 × 2 × √3`
= `4 + 3 - 4√3`
= 7` - 4√3`
`x^2 + 1/x^2`
= `(7 + 4√3) + (7 - 4√3)`
=` 7 + 4√3 + 7 - 4√3`
=` 7 + 7 + 4√3 - 4√3`
= `14`