How do you evaluate #3\sqrt { 3} ( 6\sqrt { 6} - 9\sqrt { 9} )#?

1 Answer
Jun 2, 2018

Answer:

See a solution process below:

Explanation:

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)#