Question

Simplify the following:

`1/(sqrt(2) + sqrt(3)) + 1/(sqrt(3) + sqrt(4)) + 1/(sqrt(5) + sqrt(6)) + 1/(sqrt(6) + sqrt(7))`

Answer 1

Given: \[ \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} \]

Stepwise calculation:

1. Rationalise each term by multiplying the numerator and denominator by the conjugate of the denominator:

`1/(sqrt(a) + sqrt(b)) xx (sqrt(b) - sqrt(a))/(sqrt(b) - sqrt(a))`

= `(sqrt(b) - sqrt(a))/((sqrt(b))^2 - (sqrt(a))^2`

= `(sqrt(b) - sqrt(a))/(b - a)`

Apply this to each term:

For `1/(sqrt(2) + sqrt(3))`:

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

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

For `1/(sqrt(3) + sqrt(4))`:

= `(sqrt(4) - sqrt(3))/(4 - 3)`

= `2 - sqrt(3)`

For `1/(sqrt(5) + sqrt(6))`:

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

= `sqrt(6) - sqrt(5)`

For `1/(sqrt(6) + sqrt(7))`:

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

= `sqrt(7) - sqrt(6)`

2. Sum all results:

`(sqrt(3) - sqrt(2)) + (2 - sqrt(3)) + (sqrt(6) - sqrt(5)) + (sqrt(7) - sqrt(6))`

3. Combine like terms carefully:

`(sqrt(3) - sqrt(3)) + (2) + (sqrt(6) - sqrt(6)) + (sqrt(7) - sqrt(5)) - sqrt(2)`

The `sqrt(3)` and `sqrt(6)` terms cancel out: 

`0 + 2 + 0 + (sqrt(7) - sqrt(5)) - sqrt(2)`

= `2 + (sqrt(7) - sqrt(5) - sqrt(2))`

So the expression simplifies to `2 + sqrt(7) - sqrt(5) - sqrt(2)`.