Simplify the denominator by remembering that
#(a - b) xx (a + b) = (a^2 - b^2)#
and therefore we can get rid of the roots in the denominator by multiplying by #3sqrt(3) + sqrt(6)#
Of course if we multiply the denominator by #3sqrt(3) + sqrt(6)# we will have to multiply the numerator by that amount as well.
#(4 + sqrt(2))/(3sqrt(3)-sqrt(6)) * (3sqrt(3)+sqrt(6))/(3sqrt(3)+sqrt(6))#
#= (4(3sqrt(3)+sqrt(6)) + sqrt(2)(3sqrt(3)+sqrt(6)))/(27 - 6)#
#= (12 sqrt(3) + 4 sqrt(6) + 3sqrt(6) + sqrt(12))/21#
#= (12sqrt(3) + 7sqrt(6) + sqrt(12))/21#
or
#sqrt(3) (12 + 7sqrt(2) + sqrt(4))/21#
#= sqrt(3) (14 + 7sqrt(2))/21#
#= 7sqrt(3) * (2 + sqrt(2))/3#
or, possibly
#= 7 (2+sqrt(2))/sqrt(3)#
