To rationalize the denominator and simplify, first, multiply the fraction by the appropriate form of #1#:
#(sqrt(5x) - sqrt(6y))/(sqrt(5x) - sqrt(6y)) xx (sqrt(x) - sqrt(y))/(sqrt(5x) + sqrt(6y)) =>#
#((sqrt(5x) - sqrt(6y))(sqrt(x) - sqrt(y)))/((sqrt(5x) - sqrt(6y))(sqrt(5x) + sqrt(6y))) =>#
#(sqrt(5x)sqrt(x) - sqrt(5x)sqrt(y) - sqrt(6y)sqrt(x) + sqrt(6y)sqrt(y))/((sqrt(5x))^2 + sqrt(5x)sqrt(6y) - sqrt(5x)sqrt(6y) - (sqrt(6y))^2) =>#
#(sqrt(5x * x) - sqrt(5x * y) - sqrt(6y * x) + sqrt(6y * y))/(5x + 0 - 6y) =>#
#(sqrt(5x^2) - sqrt(5xy) - sqrt(6xy) + sqrt(6y^2))/(5x - 6y) =>#
#(sqrt(5)x - sqrt(5xy) - sqrt(6xy) + sqrt(6)y)/(5x - 6y)#
If required, we can factor the numerator as:
#(sqrt(5)x - sqrt(5)sqrt(xy) - sqrt(6)sqrt(xy) + sqrt(6)y)/(5x - 6y) =>#
#(sqrt(5)(x - sqrt(xy)) - sqrt(6)(sqrt(xy) - y))/(5x - 6y)#
Or, we can also factor the numerator as:
#(sqrt(5)x - sqrt(5)sqrt(xy) - sqrt(6)sqrt(xy) + sqrt(6)y)/(5x - 6y) =>#
#(sqrt(5)x - sqrt(xy)(sqrt(5) + sqrt(6)) + sqrt(6)y)/(5x - 6y)#
