Answer
$\displaystyle \sqrt{\frac{18}{11}}=\frac{3\sqrt{22}}{11}$
Work Step by Step
$\sqrt{\frac{18}{11}}\qquad$ ...apply the Quotient Property:$\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
$=\displaystyle \frac{\sqrt{18}}{\sqrt{11}}\qquad$ ...rewrite 18 as a product of two factors so that one factor is a perfect square. ($18=9\cdot 2$)
$=\displaystyle \frac{\sqrt{9\cdot 2}}{\sqrt{11}}\qquad$ ...use the Product Property of square roots in the numerator:$\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$
$=\displaystyle \frac{\sqrt{9}\cdot\sqrt{2}}{\sqrt{11}}\qquad$ ...evaluate part of the numerator ($\sqrt{9}=3$)
$=\displaystyle \frac{3\sqrt{2}}{\sqrt{11}}\qquad$ ...rationalize the denominator by multyplying both the numerator and the denominator with $\sqrt{11}$.
$=\displaystyle \frac{3\sqrt{2}\cdot\sqrt{11}}{\sqrt{11}\cdot\sqrt{11}}\qquad$ ...use the Product Property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$
$=\displaystyle \frac{3\sqrt{2\cdot 11}}{\sqrt{11\cdot 11}}\qquad$ ...simplify.($\sqrt{11\cdot 11}=11$).
$=\displaystyle \frac{3\sqrt{22}}{11}$
