ORCCA Rationalizing the Denominator

Section 6.2 Rationalizing the Denominator

Objectives: PCC Course Content and Outcome Guide

MTH 65 CCOG 2.f

A radical expression typically has several equivalent forms. For example,

\frac{\sqrt{2}}{3}

and

\frac{2}{\sqrt{6}}

are the same number. Mathematics has a preference for one of these forms over the other, and this section is about how to convert a given radical expression to that form.

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Figure 6.2.1. Alternative Video Lesson

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Subsection 6.2.1 Rationalizing the Denominator

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To simplify radical expressions, we have seen that it helps to make the radicand as small as possible. Another helpful principle is to not leave any irrational numbers, such as

\sqrt{3}

2 \sqrt{5},

in the denominator of a fraction. In other words, we want the denominator to be rational. The process of dealing with such numbers in the denominator is called rationalizing the denominator.

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Let’s see how we can replace

\frac{1}{\sqrt{5}}

with an equivalent expression that has no radical expressions in its denominator. If we multiply a radical by itself, the result is the radicand, by Definition 6.1.2. As an example:

\sqrt{5} \cdot \sqrt{5} = 5

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With

\frac{1}{\sqrt{5}},

we may multiply both the numerator and denominator by the same nonzero number and have an equivalent expression. If we multiply the numerator and denominator by

\sqrt{5},

we have:

\begin{aligned} \frac{1}{\sqrt{5}} & = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \\ = \frac{1 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} \\ = \frac{\sqrt{5}}{5} \end{aligned}

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And voilà, we have an expression with no radical in its denominator. We can use a calculator to verify that

\frac{1}{\sqrt{5}} \approx 0.4472,

and also

\frac{\sqrt{5}}{5} \approx 0.4472 .

They are equal.

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Example 6.2.2.

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Rationalize the denominator of the expressions.

$\frac{3}{\sqrt{6}}$
$\frac{\sqrt{5}}{\sqrt{72}}$

Explanation.

To rationalize the denominator of $\frac{3}{\sqrt{6}},$ we multiply both the numerator and denominator by $\frac{\sqrt{6}}{\sqrt{6}} .$

$\begin{aligned} \frac{3}{\sqrt{6}} & = \frac{3}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} \\ = \frac{3 \sqrt{6}}{6} \\ = \frac{\sqrt{6}}{2} \end{aligned}$

Note that we reduced a fraction $\frac{3}{6}$ whose numerator and denominator were no longer inside the radical.
To rationalize the denominator of $\frac{\sqrt{5}}{\sqrt{72}},$ we could multiply both the numerator and denominator by $\sqrt{72},$ and it would be effective; however, we should note that the $\sqrt{72}$ in the denominator can be reduced first. Doing this will simplify the arithmetic because there will be smaller numbers to work with.

$\begin{aligned} \frac{\sqrt{5}}{\sqrt{72}} & = \frac{\sqrt{5}}{\sqrt{36 \cdot 2}} \\ = \frac{\sqrt{5}}{\sqrt{36} \cdot \sqrt{2}} \\ = \frac{\sqrt{5}}{6 \cdot \sqrt{2}} \end{aligned}$
Now all that remains is to multiply the numerator and denominator by $\sqrt{2} .$
$\begin{aligned} = \frac{\sqrt{5}}{6 \cdot \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \\ = \frac{\sqrt{10}}{6 \cdot 2} \\ = \frac{\sqrt{10}}{12} \end{aligned}$

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Checkpoint 6.2.3.

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Rationalize the denominator in

\frac{2}{\sqrt{10}} .

Explanation.

We will rationalize the denominator by multiplying the numerator and denominator by

\sqrt{10} :

\begin{aligned} \frac{2}{\sqrt{10}} & = \frac{2}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} \\ = \frac{2 \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} \\ = \frac{2 \sqrt{10}}{10} \\ = \frac{\sqrt{10}}{5} \end{aligned}

Again note that the fraction was simplified in the last step.

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Example 6.2.4.

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Rationalize the denominator in

\sqrt{\frac{2}{7}} .

Explanation.

This example is slightly different. The entire fraction, including its denominator, is within a radical. Having a denominator within a radical is just as undesirable as having a radical in a denominator. So we want to do something to change the expression.