Use this rule of radicals to rewrite the expression:
sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))√a⋅√b=√a⋅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))62+√12⇒62+√4⋅3⇒62+(√4⋅√3)⇒62+2√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))) =>62+2√3⇒2×3(2×1)+(2×√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)
