Question

Simplify `1/(3 - sqrt(8)) - 1/(sqrt(8) - sqrt(7)) + 1/(sqrt(7) - sqrt(6)) - 1/(sqrt(6) - sqrt(5)) + 1/(sqrt(5) - 2)`

Answer 1

Let P = `1/(3 - sqrt(8)) - 1/(sqrt(8) - sqrt(7)) + 1/(sqrt(7) - sqrt(6)) - 1/(sqrt(6) - sqrt(5)) + 1/(sqrt(5) - 2)`   ......(1)

`1/(3 - sqrt(8)) = 1/(3 - sqrt(8)) xx (3 + sqrt(8))/(3 + sqrt(8))`

= `(3 + sqrt(8))/(3^2 - (sqrt(8))^2`

`1/(3 - sqrt(8)) = (3 + sqrt(8))/(9 - 8)`

= `3 + sqrt(8)`  ......(2)

`1/(sqrt(8) - sqrt(7)) = 1/(sqrt(8) - sqrt(7)) xx  (sqrt(8) + sqrt(7))/(sqrt(8) + sqrt(7))`

= `(sqrt(8) + sqrt(7))/((sqrt(8))^2 - (sqrt(7))^2`

`1/(sqrt(8) - sqrt(7)) = (sqrt(8) + sqrt(7))/(8 - 7)`

= `sqrt(8) + sqrt(7)`  .....(3)

`1/(sqrt(7) - sqrt(6)) = 1/(sqrt(7) - sqrt(6)) xx (sqrt(7) + sqrt(6))/(sqrt(7) + sqrt(6))`

= `(sqrt(7) + sqrt(6))/((sqrt(7))^2 - (sqrt(6))^2`

`1/(sqrt(7) - sqrt(6)) = (sqrt(7) + sqrt(6))/(7 - 6)`

= `sqrt(7) + sqrt(6)`  ......(4)

`1/(sqrt(6) - sqrt(5)) = 1/(sqrt(6) - sqrt(5)) xx (sqrt(6) + sqrt(5))/(sqrt(6) + sqrt(5))`

= `(sqrt(6) + sqrt(5))/((sqrt(6))^2 - (sqrt(5))^2`

`1/(sqrt(6) - sqrt(5)) = (sqrt(6) + sqrt(5))/(6 - 5)`

= `sqrt(6) + sqrt(5)`  ......(5)

`1/(sqrt(5) - 2) = 1/(sqrt(5) - 2) xx (sqrt(5) + 2)/(sqrt(5) + 2)`

= `(sqrt(5) + 2)/((sqrt(5))^2 - 2^2`

`1/(sqrt(5) - 2) = (sqrt(5) + 2)/(5 - 4)`

= `sqrt(5) + 2`  ......(6)

Using equations (2), (3), (4), (5) and (6)

By Equation (1)

P = `(3 + sqrt(8)) - (sqrt(8) + sqrt(7)) + (sqrt(7) + sqrt(6)) - (sqrt(6) + sqrt(5)) + (sqrt(5) + 2)`

P = `3 + sqrt(8) - sqrt(8) - sqrt(7) + sqrt(7) + sqrt(6) - sqrt(6) - sqrt(5) + sqrt(5) + 2`

P = 3 + 2

= 5