Question

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

Answer 1

See a solution process below:

Explanation:

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)`