How do you multiply #3\sqrt { 15n } ( - 4\sqrt { 10n } + 4)#?

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Answer 1 · 2017-07-19T11:35:54

See a solution process below:

First, multiply each term within the parenthesis by the term outside the parenthesis:

#color(red)(3sqrt(15n))(-4sqrt(10n) + 4) =>#

#(color(red)(3sqrt(15n)) * -4sqrt(10n)) + (color(red)(3sqrt(15n)) * 4) =>#

#(color(red)(3) * -4 * color(red)(sqrt(15n))sqrt(10n)) + (4 * color(red)(3sqrt(15n))) =>#

#-12sqrt(15n)sqrt(10n) + 12sqrt(15n)#

Next, we can use this rule for radicals to multiply the radicals on the left side of the expression:

#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#

#-12sqrt(15n)sqrt(10n) + 12sqrt(15n) =>#

#-12sqrt(15n * 10n) + 12sqrt(15n) =>#

#-12sqrt(150n^2) + 12sqrt(15n)#

Then, we can rewrite the radical on the left as:

#-12sqrt(25n^2 * 6) + 12sqrt(15n)#

Now, we can use this rule of radicals (the reverse of the above rule to complete the multiplication and simplification:

#sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))#

#-12sqrt(25n^2 * 6) + 12sqrt(15n) =>#

#-12sqrt(25n^2)sqrt(6) + 12sqrt(15n) =>#

#(-12 * 5nsqrt(6)) + 12sqrt(15n) =>#

#-60nsqrt(6) + 12sqrt(15n)#