If `x = sqrt7 + sqrt6`, then find the simpliest value of (a) `x - (1)/(x)`, (b) `x + (1)/(x)`, (c) `x^(2) + (1)/(x^(2))`, (d) `x^(3) + (1)/(x^(3))`

Explanation

Given that `x = sqrt7 + sqrt6 " " :. (1)/(x) = (1)/(sqrt7 + sqrt6)`
or, `(1)/(x) = (1)/(sqrt7 + sqrt6) xx (sqrt7 - sqrt6)/(sqrt7 + sqrt6) = (sqrt7 - sqrt6)/((sqrt7)^(2) - (sqrt6)^(2)) = (sqrt7 - sqrt6)/(7 - 6)`
`= sqrt7 - sqrt6`.
`:.` (a) `x - (1)/(x) = sqrt7 + sqrt6 - (sqrt7 - sqrt6) = sqrt7 + sqrt6 - sqrt7 + sqrt6 = 2 sqrt6`
(b) `x + (1)/(x) = sqrt7 + sqrt6 + sqrt7 - sqrt6 = 2 sqrt7`
(c) `x^(2) + (1)/(x^(2)) = (x)^(2) + ((1)/(x))^(2) = (x + (1)/(x))^(2) -2.x. (1)/(x)`
`= (2 sqrt7)^(2) -2`
`= 28 -2 = 26`
(d) `x^(3) + (1)/(x^(3)) = (x + (1)/(x))^(3) -3.x. (1)/(x) (x + (1)/(x))`
`= (2 sqrt7)^(3) -3 xx 2 sqrt7`
`= 56 sqrt7 - 6sqrt7 = 50 sqrt7. " " [ :' (2 sqrt7)^(3) = 2 xx 2 xx 2 xx sqrt7 xx sqrt7 xx sqrt7 = 8 xx 7 xx sqrt7 = 56 sqrt7]`