First, expand the term in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis and then use this rule for radicals to combine terms:
sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))
color(red)(3sqrt(3))(6sqrt(6) - 9sqrt(9)) =>
(color(red)(3sqrt(3)) * 6sqrt(6)) - (color(red)(3sqrt(3)) * 9sqrt(9)) =>
(color(red)(3) * 6 * color(red)(sqrt(3)) * sqrt(6)) - (color(red)(3) * 9 * color(red)(sqrt(3)) * sqrt(9)) =>
(18 * sqrt(color(red)(3) * 6)) - (27 * sqrt(color(red)(3) * 9)) =>
18sqrt(18) - 27sqrt(27)
Next, rewrite the radicals as:
18sqrt(9 * 2) - 27sqrt(9 * 3)
Use this rule for radicals to simplify the radicals:
sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))
(18sqrt(9) * sqrt(2)) - (27sqrt(9)sqrt(3)) =>
(18 * 3 * sqrt(2)) - (27 * 3 * sqrt(3)) =>
54sqrt(2) - 81sqrt(3)
