Question

Simplify the following:

`1/(sqrt(2) + 1) + 1/(sqrt(3) + sqrt(2)) + 3/(sqrt(6) + sqrt(3)) - 2/(sqrt(6) + 2)`

Answer 1

Given expression: `1/(sqrt(2) + 1) + 1/(sqrt(3) + sqrt(2)) + 3/(sqrt(6) + sqrt(3)) - 2/(sqrt(6) + 2)`

Step 1: Rationalize each denominator individually.

1. `1/(sqrt(2) + 1) : 1/(sqrt(2) + 1) xx (sqrt(2) - 1)/(sqrt(2) - 1)` 

= `(sqrt(2) - 1)/(2 - 1)`

= `sqrt(2) - 1`

2. `1/(sqrt(3) + sqrt(2)): 1/(sqrt(3) + sqrt(2)) xx (sqrt(3) - sqrt(2))/(sqrt(3) - sqrt(2))`

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

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

3. `3/(sqrt(6) + sqrt(3)): 3/(sqrt(6) + sqrt(3)) xx (sqrt(6) - sqrt(3))/(sqrt(6) - sqrt(3))`

= `(3(sqrt(6) - sqrt(3)))/(6 - 3)`

= `(3(sqrt(6) - sqrt(3)))/3`

= `sqrt(6) - sqrt(3)`

4. `2/(sqrt(6) + 2): 2/(sqrt(6) + 2) xx (sqrt(6) - 2)/(sqrt(6) - 2)`

= `(2(sqrt(6) - 2))/(6 - 4)`

= `(2(sqrt(6) - 2))/2`

= `sqrt(6) - 2`

Step 2: Substitute simplified terms back into the expression:

`(sqrt(2) - 1) + (sqrt(3) - sqrt(2)) + (sqrt(6) - sqrt(3)) - (sqrt(6) - 2)`

Step 3: Simplify by combining like terms:

`sqrt(2) - 1 + sqrt(3) - sqrt(2) + sqrt(6) - sqrt(3) - sqrt(6) + 2`

Grouping and cancelling terms:

`sqrt(2)` and `-sqrt(2)` cancel.

`sqrt(3)` and `-sqrt(3)` cancel.

`sqrt(6)` and `-sqrt(6)` cancel.

Remaining terms: –1 + 2 = 1

The given expression simplifies to 1.