How do you simplify #6/(2+sqrt12)#?

1 Answer
Jun 11, 2017

Answer:

See a solution process below:

Explanation:

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)#