Question

If x = `(4sqrt(6))/(sqrt(2) + sqrt(3)` find the value of `(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`

Answer 1

x = `(4sqrt(6))/(sqrt(2) + sqrt(3)`

⇒ `(4sqrt(2) xx sqrt(3))/(sqrt(2) + sqrt(3)`

`x/(2sqrt(2)) = (2sqrt(3))/(sqrt(2) + sqrt(3)`
Applying componendo and dividendo,

`(x + 2sqrt(2))/(x - 2sqrt(2))`

= `(2sqrt(3) + sqrt(2) + sqrt(3))/(2sqrt(3) - sqrt(2) - sqrt(3))`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)`               ...(i)

Again `x/(2sqrt(3)) = (2sqrt(2))/(sqrt(2) + sqrt(3)`
Applying componendo and dividendo,

`(x + 2sqrt(3))/(x - 2sqrt(3))`

= `(2sqrt(2) + sqrt(2) + sqrt(3))/(2sqrt(2) - sqrt(2) - sqrt(3))`

= `(3sqrt(2) + sqrt(3))/(sqrt(2) - sqrt(3)`             ...(ii)
Adding (i) and (ii)

`(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (3sqrt(2) + sqrt(3))/(sqrt(2) - sqrt(3)`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) - (3sqrt(2) + sqrt(3))/(sqrt(3) - sqrt(2)`

= `(3sqrt(3) + sqrt(2) - 3sqrt(2) - sqrt(3))/(sqrt(3) - sqrt(2)`

= `(2sqrt(3) - 2sqrt(2))/(sqrt(3) - sqrt(2)`

= `(2(sqrt(3) - sqrt(2)))/(sqrt(3) - sqrt(2)`
= 2.