9.5 Divide Square Roots

$\begin{array}{cccc}& & & \frac{\sqrt{54}}{6}\hfill \\ \text{Simplify the radical.}\hfill & & & \frac{\sqrt{9}·\sqrt{6}}{6}\hfill \\ \text{Simplify.}\hfill & & & \frac{3\sqrt{6}}{6}\hfill \\ \text{Remove the common factors.}\hfill & & & \frac{\cancel{3}\sqrt{6}}{\cancel{3}·2}\hfill \\ \text{Simplify.}\hfill & & & \frac{\sqrt{6}}{2}\hfill \end{array}$

$\begin{array}{cccc}& & & \frac{6-\sqrt{24}}{12}\hfill \\ \text{Simplify the radical.}\hfill & & & \frac{6-\sqrt{4}·\sqrt{6}}{12}\hfill \\ \text{Simplify.}\hfill & & & \frac{6-2\sqrt{6}}{12}\hfill \\ \begin{array}{c}\text{Factor the common factor from the}\hfill \\ \text{numerator.}\hfill \end{array}\hfill & & & \frac{2\left(3-\sqrt{6}\right)}{2·6}\hfill \\ \text{Remove the common factors.}\hfill & & & \frac{\cancel{2}\left(3-\sqrt{6}\right)}{\cancel{2}·6}\hfill \\ \text{Simplify.}\hfill & & & \frac{3-\sqrt{6}}{6}\hfill \end{array}$

$\begin{array}{cccc}& & & \frac{\sqrt{27}}{\sqrt{75}}\hfill \\ \\ \\ \begin{array}{c}\text{Neither radicand is a perfect square, so}\hfill \\ \text{rewrite using the quotient property of}\hfill \\ \text{square roots.}\hfill \end{array}\hfill & & & \sqrt{\frac{27}{75}}\hfill \\ \\ \\ \begin{array}{c}\text{Remove common factors in the numerator}\hfill \\ \text{and denominator.}\hfill \end{array}\hfill & & & \sqrt{\frac{\cancel{3}·9}{\cancel{3}·25}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \sqrt{\frac{9}{25}}\hfill \\ & & & \frac{3}{5}\hfill \end{array}$

$\begin{array}{cccc}& & & \frac{\sqrt{6y^5}}{\sqrt{2y}}\hfill \\ \\ \\ \begin{array}{c}\text{Neither radicand is a perfect square, so}\hfill \\ \text{rewrite using the quotient property of}\hfill \\ \text{square roots.}\hfill \end{array}\hfill & & & \sqrt{\frac{6y^5}{2y}}\hfill \\ \\ \\ \begin{array}{c}\text{Remove common factors in the numerator}\hfill \\ \text{and denominator.}\hfill \end{array}\hfill & & & \sqrt{\frac{\cancel{2}·3·y^4·\cancel{y}}{\cancel{2}·\cancel{y}}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \sqrt{3y^4}\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & y^2\sqrt{3}\hfill \end{array}$

$\begin{array}{cccc}& & & \frac{\sqrt{72x^3}}{\sqrt{162x}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite using the quotient property of}\hfill \\ \text{square roots.}\hfill \end{array}\hfill & & & \sqrt{\frac{72x^3}{162x}}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \sqrt{\frac{\cancel{18}·4·x^2·\cancel{x}}{\cancel{18}·9·\cancel{x}}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \sqrt{\frac{4x^2}{9}}\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \frac{2x}{3}\hfill \end{array}$

$\begin{array}{cccc}& & & \frac{\sqrt{147a{b}^8}}{\sqrt{3a^3{b}^4}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite using the quotient property of}\hfill \\ \text{square roots.}\hfill \end{array}\hfill & & & \sqrt{\frac{147a{b}^8}{3a^3{b}^4}}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \sqrt{\frac{49b^4}{a^2}}\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \frac{7b^2}{a}\hfill \end{array}$

To rationalize a denominator, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

$\begin{array}{cccc}& & & \frac{4}{\sqrt{3}}\hfill \\ \\ \\ \text{Multiply both the numerator and denominator by}\sqrt{3}.\hfill & & & \frac{4·\sqrt{3}}{\sqrt{3}·\sqrt{3}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \frac{4\sqrt{3}}{3}\hfill \end{array}$

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by $\sqrt{6}$.

Multiply both the numerator and the denominator by $\sqrt{6}$.
Simplify.
Remove common factors.
Simplify.