Question

How do you simplify `(sqrt(6)-2)/(11+sqrt(6))`?

Answer 1

`(13sqrt(6)-28)/115`

Explanation:

`(sqrt(6) -2)/(11+sqrt(6))`
`=(sqrt(6) -2)/(11+sqrt(6)) * (11-sqrt(6))/(11-sqrt(6))`

this is a trick to get rid of the `sqrt(6)` of the `(11+sqrt(6))` in the denominator. We are basically constructing what is called the "conjugate" of the denominator, `(11-sqrt(6))`, which is exactly the same thing as the denominator but with the opposite sign in the middle. If the denominator is `(a+b)`, the conjugate is `(a-b)`. If the denominator is `(a-b)`, the conjugate is `(a+b)`.

Why do we want to multiply by the conjugate?
because:
`(a+b)*(a-b)=a^2-b^2`
so if a or b (or both) were square-roots, the results is squared and we are thereby getting rid of the square-root.

Note that we need to multiply the top (numerator) also by the same conjugate-of-the-denominator so that
`(11-sqrt(6))/(11-sqrt(6)) = 1` (as long as it's not `0/0` it is fine).
that is, we're only multiplying by 1 so we are not affecting the results at all.

Then it becomes easy:
`=((sqrt(6) - 2)(11-sqrt(6))) / (11^2-6)`
`=(11sqrt(6)-sqrt(6)sqrt(6)-22+2sqrt(6))/(121-6)`
`=(13sqrt(6)-28)/115`

At this stage, I don't see any more ways to simplify this further, so this must be the answer.