Skip to main content
permalink

Section 6.2 Rationalizing the Denominator

permalink

permalinkA radical expression typically has several equivalent forms. For example, 23 and 26 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.

permalink
Figure 6.2.1. Alternative Video Lesson
permalink

Subsection 6.2.1 Rationalizing the Denominator

permalinkTo 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 3 or 25, 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.

permalinkLet's see how we can replace 15 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:

5β‹…5=5

permalinkWith 15, we may multiply both the numerator and denominator by the same non-zero number and have an equivalent expression. If we multiply the numerator and denominator by 5, we have:

15=15β‹…55=1β‹…55β‹…5=55

permalinkAnd voilΓ , we have an expression with no radical in its denominator. We can use a calculator to verify that 15β‰ˆ0.4472, and also 55β‰ˆ0.4472. They are equal.

permalink
Example 6.2.2.

Rationalize the denominator of the expressions.

  1. 36

  2. 572

Explanation
  1. To rationalize the denominator of \(\frac{3}{\sqrt{6}}\text{,}\) we multiply both the numerator and denominator by \(\frac{\sqrt{6}}{\sqrt{6}}\text{.}\)

    \begin{align*} \frac{3}{\sqrt{6}}\amp=\frac{3}{\sqrt{6}}\multiplyright{\frac{\sqrt{6}}{\sqrt{6}}}\\ \amp=\frac{3\sqrt{6}}{6}\\ \amp=\frac{\sqrt{6}}{2} \end{align*}

    Note that we reduced a fraction \(\frac{3}{6}\) whose numerator and denominator were no longer inside the radical.

  2. To rationalize the denominator of \(\frac{\sqrt{5}}{\sqrt{72}}\text{,}\) we could 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. Doing this will simplify the arithmetic because there will be smaller numbers to work with.

    \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 2}\\ \amp=\frac{\sqrt{10}}{12} \end{align*}
permalink
Checkpoint 6.2.3.
permalink
Example 6.2.4.

Rationalize the denominator in 27.

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.

\begin{align*} \sqrt{\frac{2}{7}}\amp=\frac{\sqrt{2}}{\sqrt{7}}\\ \amp=\frac{\sqrt{2}}{\sqrt{7}}\multiplyright{\frac{\sqrt{7}}{\sqrt{7}}}\\ \amp=\frac{\sqrt{2}\cdot\sqrt{7}}{\sqrt{7}\cdot\sqrt{7}}\\ \amp=\frac{\sqrt{14}}{7} \end{align*}
permalink

Subsection 6.2.2 Rationalize the Denominator Using the Difference of Squares Formula

permalinkConside the number 12+1. Its denominator is irrational, approximately 2.414…. Can we rewrite this as an equivalent expression where the denominator is rational? Let's try multiplying the numerator and denominator by 2:

12+1=1(2+1)β‹…22=22β‹…2+1β‹…2=22+2

permalinkWe removed one radical from the denominator, but created another. We need to find another method. The difference of squares formula will help:

(a+b)(aβˆ’b)=a2βˆ’b2

permalinkThose two squares in a2βˆ’b2 can be used as a tool to annihilate radicals. Take 12+1, and multiply both the numerator and denominator by 2βˆ’1:

12+1=1(2+1)β‹…(2βˆ’1)(2βˆ’1)=2βˆ’1(2)2βˆ’(1)2=2βˆ’12βˆ’1=2βˆ’11=2βˆ’1
permalink
Example 6.2.5.

Rationalize the denominator in 7βˆ’25+3.

Explanation

To address the radicals in the denominator, we multiply both numerator and denominator by \(\sqrt{5}-\sqrt{3}\text{.}\)

\begin{align*} \frac{\sqrt{7}-\sqrt{2}}{\sqrt{5}+\sqrt{3}}\amp=\frac{\sqrt{7}-\sqrt{2}}{\sqrt{5}+\sqrt{3}}\multiplyright{\frac{\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)}}\\ \amp=\frac{\sqrt{7}\multiplyright{\sqrt{5}}-\sqrt{7}\multiplyright{\sqrt{3}}-\sqrt{2}\multiplyright{\sqrt{5}}-\sqrt{2}\multiplyright{-\sqrt{3}}}{\left(\sqrt{5}\right)^2-\left(\sqrt{3}\right)^2}\\ \amp=\frac{\sqrt{35}-\sqrt{21}-\sqrt{10}+\sqrt{6}}{5-3}\\ \amp=\frac{\sqrt{35}-\sqrt{21}-\sqrt{10}+\sqrt{6}}{2} \end{align*}
permalink
Checkpoint 6.2.6.
permalink

Reading Questions 6.2.3 Reading Questions

permalink
1.

To rationalize a denominator in an expression like 35, explain the first step you will take.

permalink
2.

What is the special pattern from Section 5.5 that helps to rationalize the denominator in an expression like 32+5?

permalink

Exercises 6.2.4 Exercises

Review and Warmup
permalink
1.

Rationalize the denominator and simplify the expression.

16=

permalink
2.

Rationalize the denominator and simplify the expression.

16=

permalink
3.

Rationalize the denominator and simplify the expression.

77=

permalink
4.

Rationalize the denominator and simplify the expression.

4010=

permalink
5.

Rationalize the denominator and simplify the expression.

1180=

permalink
6.

Rationalize the denominator and simplify the expression.

18=

permalink
7.

Rationalize the denominator and simplify the expression.

2252=

permalink
8.

Rationalize the denominator and simplify the expression.

9180=

Rationalizing the Denominator
permalink
9.

Evaluate the following.

34 = .

permalink
10.

Evaluate the following.

564 = .

permalink
11.

Rationalize the denominator and simplify the expression.

16=

permalink
12.

Rationalize the denominator and simplify the expression.

17=

permalink
13.

Rationalize the denominator and simplify the expression.

710=

permalink
14.

Rationalize the denominator and simplify the expression.

710=

permalink
15.

Rationalize the denominator and simplify the expression.

582=

permalink
16.

Rationalize the denominator and simplify the expression.

733=

permalink
17.

Rationalize the denominator and simplify the expression.

610=

permalink
18.

Rationalize the denominator and simplify the expression.

2014=

permalink
19.

Rationalize the denominator and simplify the expression.

186=

permalink
20.

Rationalize the denominator and simplify the expression.

126=

permalink
21.

Rationalize the denominator and simplify the expression.

1175=

permalink
22.

Rationalize the denominator and simplify the expression.

1180=

permalink
23.

Rationalize the denominator and simplify the expression.

272=

permalink
24.

Rationalize the denominator and simplify the expression.

632=

permalink
25.

Rationalize the denominator and simplify the expression.

79=

permalink
26.

Rationalize the denominator and simplify the expression.

316=

permalink
27.

Rationalize the denominator and simplify the expression.

92=

permalink
28.

Rationalize the denominator and simplify the expression.

812=

permalink
29.

Rationalize the denominator and simplify the expression.

112=

permalink
30.

Rationalize the denominator and simplify the expression.

1315=

permalink
31.

Rationalize the denominator and simplify the expression.

1087=

permalink
32.

Rationalize the denominator and simplify the expression.

725=

permalink
33.

Rationalize the denominator and simplify the expression.

4x=

permalink
34.

Rationalize the denominator and simplify the expression.

2y=

permalink
35.

Rationalize the denominator and simplify the expression.

52=

permalink
36.

Rationalize the denominator and simplify the expression.

611=

permalink
37.

Rationalize the denominator and simplify the expression.

1148=

permalink
38.

Rationalize the denominator and simplify the expression.

11175=

Rationalizing the Denominator Using the Difference of Squares Formula
permalink
39.

Rationalize the denominator and simplify the expression.

715+7=

permalink
40.

Rationalize the denominator and simplify the expression.

222+5=

permalink
41.

Rationalize the denominator and simplify the expression.

822+9=

permalink
42.

Rationalize the denominator and simplify the expression.

214+9=

permalink
43.

Rationalize the denominator and simplify the expression.

2βˆ’611+4=

permalink
44.

Rationalize the denominator and simplify the expression.

5βˆ’813+10=

permalink
45.

Rationalize the denominator and simplify the expression.

3βˆ’97+8=

permalink
46.

Rationalize the denominator and simplify the expression.

2βˆ’1011+5=