Example14.3.2
Rationalize the denominator of the expressions.
\(\frac{3}{\sqrt{6}}\)
\(\frac{\sqrt{5}}{\sqrt{72}}\)
Solution
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To rationalize the denominator of \(\frac{3}{\sqrt{6}}\text{,}\) we take the expression and multiply by a special version of \(\highlight{1}\) to make the radical in the denominator cancel.
\begin{align*} \frac{3}{\sqrt{6}}\amp=\frac{3}{\sqrt{6}}\multiplyright{\frac{\sqrt{6}}{\sqrt{6}}}\\ \amp=\frac{3\sqrt{6}}{\sqrt{36}}\\ \amp=\frac{3\sqrt{6}}{6}\\ \amp=\frac{\sqrt{6}}{2} \end{align*} -
Rationalizing the denominator of \(\frac{\sqrt{5}}{\sqrt{72}}\) is slightly trickier. We could go the brute force method and multiply both the numerator and denominator by \(\sqrt{72}\text{,}\) and it would be effective; however, we should note that the \(\sqrt{72}\) in the denominator can be reduced first. This will simplify future algebra.
\begin{align*} \frac{\sqrt{5}}{\sqrt{72}}\amp=\frac{\sqrt{5}}{\sqrt{36\cdot 2}}\\ \amp=\frac{\sqrt{5}}{\sqrt{36}\cdot\sqrt{2}}\\ \amp=\frac{\sqrt{5}}{6\cdot\sqrt{2}}\\ \end{align*}Now all that remains is to multiply the numerator and denominator by \(\sqrt{2}\text{.}\)
\begin{align*} \amp=\frac{\sqrt{5}}{6\cdot\sqrt{2}}\multiplyright{\frac{\sqrt{2}}{\sqrt{2}}}\\ \amp=\frac{\sqrt{10}}{6\cdot\sqrt{4}}\\ \amp=\frac{\sqrt{10}}{6\cdot 2}\\ \amp=\frac{\sqrt{10}}{12} \end{align*}