How do you simplify #sqrt5(3+sqrt15)#?

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Answer 1 · 2017-06-03T22:05:43

See a solution process below:

First, eliminate the parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis:

#color(red)(sqrt(5))(3 + sqrt(15)) =>#

#(color(red)(sqrt(5)) * 3) + (color(red)(sqrt(5)) * sqrt(15)) =>#

#3sqrt(5) + (sqrt(5) * sqrt(15))#

Next, use this rule for multiplying radicals to rewrite the term on the right:

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

#3sqrt(5) + (sqrt(color(red)(5)) * sqrt(color(blue)(15))) =>#

#3sqrt(5) + (sqrt(color(red)(5) * color(blue)(15))) =>#

#3sqrt(5) + sqrt(75)#

Then, we can use the same rule in reverse to again rewrite and simplify the term on the right:

#3sqrt(5) + sqrt(75) =>#

#3sqrt(5) + sqrt(color(red)(25) * color(blue)(3)) =>#

#3sqrt(5) + (sqrt(color(red)(25)) * sqrt(color(blue)(3))) =>#

#3sqrt(5) + (5 * sqrt(color(blue)(3))) =>#

#3sqrt(5) + 5sqrt(3)#