How do you rationalize the denominator and simplify ##(sqrt(11)-sqrt(2))/(sqrt(11)+sqrt(2))##?

##(13 -2sqrt(22))/9##

Multiply the numerator and the denominator by the reciprocal of the denominator

##sqrt(11) - sqrt(2)##

This will give you

##(sqrt11 - sqrt2)/(sqrt11 + sqrt2) = (sqrt11 - sqrt2)/(sqrt11 + sqrt2) xx (sqrt11 - sqrt2)/(sqrt11 - sqrt2)##

Then multiply the numerator to each other, same with the denominator to remove the radical sign. Because multiplying the same radicals the result was still the radicand itself e.

##sqrt 11 xx sqrt 11 = sqrt(11^2) = 121##

The radical expression now become

##(sqrt11 - sqrt2)/(sqrt11 + sqrt2) xx (sqrt11 - sqrt2)/(sqrt11 - sqrt2) = (sqrt11 - sqrt2)^2/(11 - 2)##

This will get you

##(11 - 2 sqrt(22) + 2)/9 = (13 -2sqrt(22))/9##