Question

If x = `sqrt3 - sqrt2`, find the value of:

(i) `x + 1/x`

(ii) `x^2 + 1/x^2`

(iii) `x^3 + 1/x^3`

(iv) `x^3 + 1/x^3 - 3(x^2 + 1/x^2) + x + 1/x`

Answer 1

x = `sqrt3 - sqrt2`

`1/x = 1/(sqrt3 - sqrt2) xx (sqrt3 + sqrt2)/(sqrt3 + sqrt2)`

`1/x = (sqrt3 + sqrt2)/((sqrt3)^2 - (sqrt2)^2)`

`1/x = (sqrt 3 + sqrt 2)`

(i) `x + 1/x`

`= (sqrt3 - sqrt2) + (sqrt3 + sqrt2)`

`= sqrt3 - cancel(sqrt2) + sqrt3 + cancel(sqrt2)`

`= sqrt3 + sqrt3`

= `2sqrt3`   

(ii) `x^2 + 1/x^2`

`= (x + 1/x)^2 - 2 * x * 1/x`    ...[a² + b² = (a + b)² - 2ab]

`= (2sqrt3)^2 - 2`

`= 4 xx 3` - 2

= 10

(iii) `x^3 + 1/x^3`

`= (x + 1/x)^3 - 3 * x * 1/x (x + 1/x)`  ...[a³ + b³ = (a + b)³ - 3 · a · b (a + b)]

`= (2sqrt3)^3 - 3 xx (2sqrt3)`

`= 8 xx 3sqrt3 - 6sqrt3`

`= 24sqrt3 - 6sqrt3`

`= 18sqrt3`

(iv) `x^3 + 1/x^3 - 3(x^2 + 1/x^2) + x + 1/x`

`= 18sqrt3 - 3(10) + 2sqrt3`

`= 20sqrt3 - 30`

`= 10(2sqrt3 - 3)`