How do you simplify the expression #sqrt3(2+3sqrt6)#?

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Answer 1 · 2017-05-13T13:50:21

See a solution process below:

First, expand the parentheses by multiplying each term within the parentheses by the factor outside the parentheses:

#color(red)(sqrt(3))(2 + 3sqrt(6)) =>#

#(color(red)(sqrt(3)) * 2) + (color(red)(sqrt(3)) * 3sqrt(6)) =>#

#2sqrt(3) + 3sqrt(3)sqrt(6)#

We can now use this rule for radicals to simplify the term on the right:

#sqrt(a) * sqrt(b) = sqrt(a * b)#

#2sqrt(3) + 3sqrt(3)sqrt(6) => #

#2sqrt(3) + 3sqrt(3 * 6) =>#

#2sqrt(3) + 3sqrt(18)#

We can use this same rule in reverse to further simplify the term on the right:

#2sqrt(3) + 3sqrt(18) =>#

#2sqrt(3) + 3sqrt(9 * 2) =>#

#2sqrt(3) + 3sqrt(9)sqrt(2) =>#

#2sqrt(3) + (3 * 3)sqrt(2) =>#

#2sqrt(3) + 9sqrt(2)#