Use this rule of radicals to rewrite the expression:
#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#
#6/(2 + sqrt(12)) => 6/(2 + sqrt(color(red)(4) * color(blue)(3))) => 6/(2 + (sqrt(color(red)(4)) * sqrt(color(blue)(3)))) => 6/(2 + 2sqrt(3))#
Now, we can factor the numerator and denominator and cancel common terms to complete the simplification:
#6/(2 + 2sqrt(3)) => (2 xx 3)/((2 xx 1) + (2 xx sqrt(3))) =>#
#(2 xx 3)/(2(1 + sqrt(3))) => (color(red)(cancel(color(black)(2))) xx 3)/(color(red)(cancel(color(black)(2)))(1 + sqrt(3))) =>#
#3/(1 + sqrt(3))#
If necessary and required we can modify this result by rationalizing the denominator or, in other words, removing all the radicals from the denominator:
#3/(1 + sqrt(3)) => (1 - sqrt(3))/(1 - sqrt(3)) xx 3/(1 + sqrt(3)) =>#
#((3 xx 1) - (3 xx sqrt(3)))/(1^2 - sqrt(3) + sqrt(3) - (sqrt(3)^2)) =>#
#(3 - 3sqrt(3))/(1 - 3) =>#
#(3 - 3sqrt(3))/-2#
Or
#-3/2(1 - sqrt(3))#
Or
#3/2(sqrt(3) - 1)#
