Answer
$\displaystyle \sqrt{\frac{19}{21}}=\frac{\sqrt{399}}{21}$
Work Step by Step
$\sqrt{\frac{19}{21}}\qquad$ ...apply the Quotient Property:$\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
$=\displaystyle \frac{\sqrt{19}}{\sqrt{21}}\qquad$ ...rationalize by multiplying both the numerator and the denominator with $\sqrt{21}$.
$=\displaystyle \frac{\sqrt{19}\cdot\sqrt{21}}{\sqrt{21}\cdot\sqrt{21}}\qquad$ ...use the Product Property of square roots:$\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$
$=\displaystyle \frac{\sqrt{19\cdot 21}}{\sqrt{21\cdot 21}}\qquad$ ...simplify.($\sqrt{21\cdot 21}=21$).
$=\displaystyle \frac{\sqrt{19\cdot 21}}{21}\qquad$ ...simplify.
$=\displaystyle \frac{\sqrt{399}}{21}\qquad$ ...we cannot simplify any further. $\sqrt{399}=\sqrt{3\cdot 7\cdot 19}$, no perfect squares.
