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

Jun 3, 2017

See a solution process below:

#### Explanation:

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

$\textcolor{red}{\sqrt{5}} \left(3 + \sqrt{15}\right) \implies$

$\left(\textcolor{red}{\sqrt{5}} \cdot 3\right) + \left(\textcolor{red}{\sqrt{5}} \cdot \sqrt{15}\right) \implies$

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

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

$\sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}}$

$3 \sqrt{5} + \left(\sqrt{\textcolor{red}{5}} \cdot \sqrt{\textcolor{b l u e}{15}}\right) \implies$

$3 \sqrt{5} + \left(\sqrt{\textcolor{red}{5} \cdot \textcolor{b l u e}{15}}\right) \implies$

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

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

$3 \sqrt{5} + \sqrt{75} \implies$

$3 \sqrt{5} + \sqrt{\textcolor{red}{25} \cdot \textcolor{b l u e}{3}} \implies$

$3 \sqrt{5} + \left(\sqrt{\textcolor{red}{25}} \cdot \sqrt{\textcolor{b l u e}{3}}\right) \implies$

$3 \sqrt{5} + \left(5 \cdot \sqrt{\textcolor{b l u e}{3}}\right) \implies$

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