How do you simplify sqrt7(sqrt14 + sqrt3)?

Oct 8, 2017

See a solution process below:

Explanation:

First, expand the terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis:

$\sqrt{\textcolor{red}{7}} \left(\sqrt{\textcolor{b l u e}{14}} + \sqrt{\textcolor{b l u e}{3}}\right) \implies$

$\left(\sqrt{\textcolor{red}{7}} \times \sqrt{\textcolor{b l u e}{14}}\right) + \left(\sqrt{\textcolor{red}{7}} \times \sqrt{\textcolor{b l u e}{3}}\right)$

Next, we can use this rule for radicals to rewrite the expression:

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

$\sqrt{\textcolor{red}{7} \times \textcolor{b l u e}{14}} + \sqrt{\textcolor{red}{7} \times \textcolor{b l u e}{3}} \implies$

$\sqrt{98} + \sqrt{21}$

We can rewrite the term on the left as:

$\sqrt{49 \times 2} + \sqrt{21}$

We can use the reverse of the rule above to simplify the term on the left:

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

$\sqrt{\textcolor{red}{49} \times \textcolor{b l u e}{2}} + \sqrt{21} \implies$

$\left(\sqrt{\textcolor{red}{49}} \times \sqrt{\textcolor{b l u e}{2}}\right) + \sqrt{21} \implies$

$\left(7 \times \sqrt{\textcolor{b l u e}{2}}\right) + \sqrt{21} \implies$

$7 \sqrt{\textcolor{b l u e}{2}} + \sqrt{21}$