How do you simplify `(sqrt 6 - sqrt 5) / (sqrt 6 + sqrt 5)`?

Your goal here is to rationalize the denominator by multiplying it by its conjugate.

The conjugate of a binomial can be determined by changing the sign of the second term. In your case, you would have

`sqrt(6) + sqrt(5) -> underbrace(sqrt(6) color(red)(-) sqrt(5))_(color(blue)("conjugate"))`

So, multiply the fraction by `1 = (sqrt(6) - sqrt(5))/(sqrt(6) - sqrt(5))` to get

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

The denominator takes the form

`color(blue)( (a-b)(a+b) = a^2 - b^2)`

and can thus be written as

`(sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5)) = (sqrt(6))^2 - (sqrt(5))^2 = 6 - 5 = 1`

The numerator takes the form

`color(blue)( (a-b)^2 = a^2 - 2ab + b^2)`

and can be written as

`(sqrt(6) - sqrt(5))(sqrt(6) - sqrt(5)) = (sqrt(6))^2 - 2sqrt(30) + (sqrt(5))^2`

`=6 - 2sqrt(30) + 5`

`= 11 - 2sqrt(30)`

The expression becomes

`((sqrt(6) - sqrt(5))(sqrt(6) - sqrt(5)))/((sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5))) = (11 - 2sqrt(30))/1 = color(green)(11 - 2sqrt(30))`