# How do you simplify 4sqrt (81 / 8)?

Jun 22, 2017

See a solution process below:

#### Explanation:

First, using this rule of radicals rewrite the expression:

sqrt(color(red)(a)/color(blue)(b)) = sqrt(color(red)(a))/sqrt(color(blue)(b)

$4 \sqrt{\frac{\textcolor{red}{81}}{\textcolor{b l u e}{8}}} \implies 4 \frac{\sqrt{\textcolor{red}{81}}}{\sqrt{\textcolor{b l u e}{8}}} \implies \frac{4 \cdot 9}{\sqrt{8}} \implies \frac{36}{\sqrt{8}}$

Next, rewrite the denominator and use this rule of exponents to simplify the denominator:

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

$\frac{36}{\sqrt{8}} \implies \frac{36}{\sqrt{\textcolor{red}{4} \cdot \textcolor{b l u e}{2}}} \implies \frac{36}{\sqrt{\textcolor{red}{4}} \cdot \sqrt{\textcolor{b l u e}{2}}} \implies \frac{36}{2 \sqrt{2}} \implies \frac{18}{\sqrt{2}}$

To rationalize the denominator or eliminate the radical from the denominator we can multiply by the appropriate form of $1$:

$\frac{18}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \implies \frac{18 \sqrt{2}}{\sqrt{2}} ^ 2 \implies \frac{18 \sqrt{2}}{2} \implies 9 \sqrt{2}$

Jun 22, 2017

$4 \sqrt{\frac{81}{8}} = 9 \sqrt{2}$

#### Explanation:

If $a , b > 0$ then:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

When simplifying square roots of fractions such as this example, I like to make the denominator into a perfect square first.

So we find:

$4 \sqrt{\frac{81}{8}} = 4 \sqrt{\frac{81 \cdot 2}{16}} = 4 \sqrt{\frac{{9}^{2} \cdot 2}{4} ^ 2} = 4 \cdot \frac{9}{4} \sqrt{2} = 9 \sqrt{2}$