# How do you rationalize the denominator and simplify sqrt (33/77)?

Mar 15, 2016

$\frac{\sqrt{21}}{7}$

#### Explanation:

$1$. Since the denominator of the fraction contains a radical, start by multiplying the numerator and denominator by $\frac{\sqrt{77}}{\sqrt{77}}$. Note that $\frac{\sqrt{77}}{\sqrt{77}} = 1$, so that value of the fraction remains the same.

$\frac{\sqrt{33}}{\sqrt{77}}$

$= \frac{\sqrt{33}}{\sqrt{77}} \left(\frac{\sqrt{77}}{\sqrt{77}}\right)$

$2$. Simplify.

$= \frac{\sqrt{33 \cdot 77}}{77}$

$= \frac{\sqrt{2541}}{77}$

$3$. Use a perfect square to break down the radical in the numerator.

$= \frac{\sqrt{{11}^{2} \cdot 21}}{77}$

$4$. Simplify.

$= \frac{11 \sqrt{21}}{77}$

$= \frac{{\textcolor{red}{\cancel{\textcolor{b l a c k}{11}}}}^{1} \sqrt{21}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{77}}}} ^ 7$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \frac{\sqrt{21}}{7} \textcolor{w h i t e}{\frac{a}{a}} |}}}$