How do you simplify $\sqrt{5} (3 + \sqrt{15})$ $sqrt5(3+sqrt15)$ ?

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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:

$\sqrt{5} (3 + \sqrt{15}) \Rightarrow$ $color(red)(sqrt(5))(3 + sqrt(15)) =>$

$(\sqrt{5} \cdot 3) + (\sqrt{5} \cdot \sqrt{15}) \Rightarrow$ $(color(red)(sqrt(5)) * 3) + (color(red)(sqrt(5)) * sqrt(15)) =>$

$3 \sqrt{5} + (\sqrt{5} \cdot \sqrt{15})$ $3sqrt(5) + (sqrt(5) * sqrt(15))$

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

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ $sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))$

$3 \sqrt{5} + (\sqrt{5} \cdot \sqrt{15}) \Rightarrow$ $3sqrt(5) + (sqrt(color(red)(5)) * sqrt(color(blue)(15))) =>$

$3 \sqrt{5} + (\sqrt{5 \cdot 15}) \Rightarrow$ $3sqrt(5) + (sqrt(color(red)(5) * color(blue)(15))) =>$

$3 \sqrt{5} + \sqrt{75}$ $3sqrt(5) + sqrt(75)$

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

$3 \sqrt{5} + \sqrt{75} \Rightarrow$ $3sqrt(5) + sqrt(75) =>$

$3 \sqrt{5} + \sqrt{25 \cdot 3} \Rightarrow$ $3sqrt(5) + sqrt(color(red)(25) * color(blue)(3)) =>$

$3 \sqrt{5} + (\sqrt{25} \cdot \sqrt{3}) \Rightarrow$ $3sqrt(5) + (sqrt(color(red)(25)) * sqrt(color(blue)(3))) =>$

$3 \sqrt{5} + (5 \cdot \sqrt{3}) \Rightarrow$ $3sqrt(5) + (5 * sqrt(color(blue)(3))) =>$

$3 \sqrt{5} + 5 \sqrt{3}$ $3sqrt(5) + 5sqrt(3)$

How do you simplify √5(3+√15)sqrt5(3+sqrt15)?