Aus Online Mathematik Brückenkurs 1

When simplifying a root expression, a common technique is to divide up the numbers under the root sign into their smallest possible integer factors and then take out the squares and see if common factor cancel each other out or can be combined together in a new way.

By successively dividing by \displaystyle 2 and \displaystyle 3, we see that

\displaystyle \begin{align} & 96=2\centerdot 48=2\centerdot 2\centerdot 24=2\centerdot 2\centerdot 2\centerdot 12=2\centerdot 2\centerdot 2\centerdot 2\centerdot 6 \\ & =2\centerdot 2\centerdot 2\centerdot 2\centerdot 2\centerdot 3=2^{5}\centerdot 3, \\ & 18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2}. \\ \end{align}

Thus,

\displaystyle \begin{align} & \sqrt{96}=\sqrt{2^{5}\centerdot 3}=\sqrt{2^{2}\centerdot 2^{2}\centerdot 2\centerdot 3}=2\centerdot 2\centerdot \sqrt{2}\centerdot \sqrt{3} \\ & \sqrt{18}=\sqrt{2\centerdot 3^{2}}=3\centerdot \sqrt{2} \\ \end{align}

and the whole quotient can be written as

\displaystyle \frac{\sqrt{96}}{\sqrt{18}}=\frac{2\centerdot 2\centerdot \sqrt{2}\centerdot \sqrt{3}}{3\centerdot \sqrt{2}}=\frac{4\sqrt{3}}{3}

NOTE: If it is difficult to work with the root sign, it is possible instead to write everything in power form

\displaystyle \begin{align} & \frac{\sqrt{96}}{\sqrt{18}}=\frac{\left( 96 \right)^{\frac{1}{2}}}{\left( 18 \right)^{{1}/{2}\;}}=\frac{\left( 2^{5}\centerdot 3 \right)^{\frac{1}{2}}}{\left( 2\centerdot 3^{2} \right)^{{1}/{2}\;}}=\frac{2^{5\centerdot \frac{1}{2}}\centerdot 3^{\frac{1}{2}}}{2^{\frac{1}{2}}\centerdot 3^{2\centerdot \frac{1}{2}}} \\ & =2^{\frac{5}{2}-\frac{1}{2}}\centerdot 3^{\frac{1}{2}-1}=2^{2}\centerdot 3^{-\frac{1}{2}}=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3} \\ \end{align}

(in the last line, we multiply top and bottom by \displaystyle \sqrt{3} ).