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:

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

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

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

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

$3sqrt(5) + 5sqrt(3)$

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