If `x = 2 + sqrt3, y = 2 - sqrt3`, then find the simpliest value of (a) `x - (1)/(x)`, (b) `y^(2) + (1)/(y^(2))`, (c) `x^(3) - (1)/(x^(3))`, (d) `xy + (1)/(xy)`

Explanation

`x = 2 + sqrt3, " " :. (1)/(x) = (1)/(2 + sqrt3) = (1)/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3)`
`= (2 - sqrt3)/((2)^(2) - (sqrt3)^(2)) = (2 - sqrt3)/(4 -3) = 2 - sqrt3`
`y = 2 - sqrt3, " " :. (1)/(y) = (1)/(2 - sqrt3) = (1)/(2 - sqrt3) xx (2 + sqrt3)/(2 + sqrt3)`
`= (2 + sqrt3)/((2)^(2) - (sqrt3)^(2)) = (2 + sqrt3)/(4 - 3) = (2 + sqrt3)/(1) = 2 + sqrt3`
`:.` (a) `x - (1)/(x) = 2 + sqrt3 - 2 + sqrt3 = 2 sqrt3`
(b) `y^(2) + (1)/(y^(2)) = (y + (1)/(y))^(2) -2.y.(1)/(y) = (2 - sqrt3 + 2 + sqrt3)^(2) -2`
`= (4)^(2) -2 = 16 - 2 = 14`
(c) `x^(3) - (1)/(x^(3)) = (x)^(3) - ((1)/(x))^(3) = (x - (1)/(x))^(3) + 3.x.(1)/(x) (x - (1)/(x))`
`= (2 sqrt3)^(3) + 3 xx 2 sqrt2 = 24 sqrt3 + 6 sqrt3 = 30 sqrt3`
(d) `xy + (1)/(xy) = (2 + sqrt3) (2 - sqrt3) + (1)/((2 + sqrt3) (2 - sqrt3))`
`= (2)^(2) - (sqrt3)^(2) + (1)/((2)^(2) - (sqrt3)^(2)) = 4 - 3 + (1)/(4 -3)`
`= 1 + (1)/(1) = 1 + 1 = 2`