Learning Objectives
By the end of this section, you will be able to:
 Divide square roots
 Rationalize a oneterm denominator
 Rationalize a twoterm denominator
Be Prepared 9.12
Before you get started, take this readiness quiz.
Find a fraction equivalent to $\frac{5}{8}$ with denominator 48.
If you missed this problem, review Example 1.64.
Be Prepared 9.13
Simplify: ${\left(\sqrt{5}\right)}^{2}$.
If you missed this problem, review Example 9.48.
Be Prepared 9.14
Multiply: $\left(7+3x\right)\left(73x\right)$.
If you missed this problem, review Example 6.54.
Divide Square Roots
We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.
Example 9.60
Simplify: $\frac{\sqrt{54}}{6}$.
Solution
$\frac{\sqrt{54}}{6}$  
Simplify the radical.  $\frac{\sqrt{9}\xb7\sqrt{6}}{6}$ 
Simplify.  $\frac{3\sqrt{6}}{6}$ 
Remove the common factors.  $\frac{\overline{)3}\sqrt{6}}{\overline{)3}\xb72}$ 
Simplify.  $\frac{\sqrt{6}}{2}$ 
Try It 9.119
Simplify: $\frac{\sqrt{32}}{8}$.
Try It 9.120
Simplify: $\frac{\sqrt{75}}{15}$.
Example 9.61
Simplify: $\frac{6\sqrt{24}}{12}$.
Solution
$\frac{6\sqrt{24}}{12}$  
Simplify the radical.  $\frac{6\sqrt{4}\xb7\sqrt{6}}{12}$ 
Simplify.  $\frac{62\sqrt{6}}{12}$ 
Factor the common factor from the numerator.  $\frac{2\left(3\sqrt{6}\right)}{2\xb76}$ 
Remove the common factors.  $\frac{\overline{)2}\left(3\sqrt{6}\right)}{\overline{)2}\xb76}$ 
Simplify.  $\frac{3\sqrt{6}}{6}$ 
Try It 9.121
Simplify: $\frac{8\sqrt{40}}{10}$.
Try It 9.122
Simplify: $\frac{10\sqrt{75}}{20}$.
We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says
Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.
We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.
Quotient Property of Square Roots
If a, b are nonnegative real numbers and $b\ne 0$, then
We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.
Example 9.62
Simplify: $\frac{\sqrt{27}}{\sqrt{75}}$.
Solution
$\frac{\sqrt{27}}{\sqrt{75}}$  
Neither radicand is a perfect square, so rewrite using the quotient property of square roots.  $\sqrt{\frac{27}{75}}$ 
Remove common factors in the numerator and denominator.  $\sqrt{\frac{\overline{)3}\xb79}{\overline{)3}\xb725}}$ 
Simplify.  $\sqrt{\frac{9}{25}}$ 
$\frac{3}{5}$ 
Try It 9.123
Simplify: $\frac{\sqrt{48}}{\sqrt{108}}$.
Try It 9.124
Simplify: $\frac{\sqrt{96}}{\sqrt{54}}$.
We will use the Quotient Property for Exponents, $\frac{{a}^{m}}{{a}^{n}}={a}^{mn}$, when we have variables with exponents in the radicands.
Example 9.63
Simplify: $\frac{\sqrt{6{y}^{5}}}{\sqrt{2y}}$.
Solution
$\frac{\sqrt{6{y}^{5}}}{\sqrt{2y}}$  
Neither radicand is a perfect square, so rewrite using the quotient property of square roots.  $\sqrt{\frac{6{y}^{5}}{2y}}$ 
Remove common factors in the numerator and denominator.  $\sqrt{\frac{\overline{)2}\xb73\xb7{y}^{4}\xb7\overline{)y}}{\overline{)2}\xb7\overline{)y}}}$ 
Simplify.  $\sqrt{3{y}^{4}}$ 
Simplify the radical.  ${y}^{2}\sqrt{3}$ 
Try It 9.125
Simplify: $\frac{\sqrt{12{r}^{3}}}{\sqrt{6r}}$.
Try It 9.126
Simplify: $\frac{\sqrt{14{p}^{9}}}{\sqrt{2{p}^{5}}}$.
Example 9.64
Simplify: $\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}$.
Solution
$\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}$  
Rewrite using the quotient property of square roots.  $\sqrt{\frac{72{x}^{3}}{162x}}$ 
Remove common factors.  $\sqrt{\frac{\overline{)18}\xb74\xb7{x}^{2}\xb7\overline{)x}}{\overline{)18}\xb79\xb7\overline{)x}}}$ 
Simplify.  $\sqrt{\frac{4{x}^{2}}{9}}$ 
Simplify the radical.  $\frac{2x}{3}$ 
Try It 9.127
Simplify: $\frac{\sqrt{50{s}^{3}}}{\sqrt{128s}}$.
Try It 9.128
Simplify: $\frac{\sqrt{75{q}^{5}}}{\sqrt{108q}}$.
Example 9.65
Simplify: $\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}$.
Solution
$\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}$  
Rewrite using the quotient property of square roots.  $\sqrt{\frac{147a{b}^{8}}{3{a}^{3}{b}^{4}}}$ 
Remove common factors.  $\sqrt{\frac{49{b}^{4}}{{a}^{2}}}$ 
Simplify the radical.  $\frac{7{b}^{2}}{a}$ 
Try It 9.129
Simplify: $\frac{\sqrt{162{x}^{10}{y}^{2}}}{\sqrt{2{x}^{6}{y}^{6}}}$.
Try It 9.130
Simplify: $\frac{\sqrt{300{m}^{3}{n}^{7}}}{\sqrt{3{m}^{5}n}}$.
Rationalize a One Term Denominator
Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. Figure 9.3 shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.
If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimalplace divisor. This was a very cumbersome process.
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.
Rationalizing the Denominator
The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.
Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.
Let’s look at a numerical example.
$\begin{array}{cccc}\text{Suppose we need an approximate value for the fraction.}\hfill & & & \frac{1}{\sqrt{2}}\hfill \\ \text{A five decimal place approximation to}\phantom{\rule{0.2em}{0ex}}\sqrt{2}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}1.41421.\hfill & & & \frac{1}{1.41421}\hfill \\ \text{Without a calculator, would you want to do this division?}\hfill & & & 1.41421\overline{)1.0}\hfill \end{array}$
But we can find a fraction equivalent to $\frac{1}{\sqrt{2}}$ by multiplying the numerator and denominator by $\sqrt{2}$.
Now if we need an approximate value, we divide $2\overline{)1.41421}$. This is much easier.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.
Similarly, a square root is not considered simplified if the radicand contains a fraction.
Simplified Square Roots
A square root is considered simplified if there are
 no perfectsquare factors in the radicand
 no fractions in the radicand
 no square roots in the denominator of a fraction
To rationalize a denominator, we use the property that ${\left(\sqrt{a}\right)}^{2}=a$. If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Example 9.66
Simplify: $\frac{4}{\sqrt{3}}$.
Solution
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.
$\frac{4}{\sqrt{3}}$  
Multiply both the numerator and denominator by $\sqrt{3}.$  $\frac{4\xb7\sqrt{3}}{\sqrt{3}\xb7\sqrt{3}}$ 
Simplify.  $\frac{4\sqrt{3}}{3}$ 
Try It 9.131
Simplify: $\frac{5}{\sqrt{3}}$.
Try It 9.132
Simplify: $\frac{6}{\sqrt{5}}$.
Example 9.67
Simplify: $\frac{8}{3\sqrt{6}}$.
Solution
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. 
Try It 9.133
Simplify: $\frac{5}{2\sqrt{5}}$.
Try It 9.134
Simplify: $\frac{9}{4\sqrt{3}}$.
Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.
Example 9.68
Simplify: $\sqrt{\frac{5}{12}}$.
Solution
The fraction is not a perfect square, so rewrite using the Quotient Property. 

Simplify the denominator  
Rationalize the denominator.  
Simplify.  
Simplify. 
Try It 9.135
Simplify: $\sqrt{\frac{7}{18}}$.
Try It 9.136
Simplify: $\sqrt{\frac{3}{32}}$.
Example 9.69
Simplify: $\sqrt{\frac{11}{28}}$.
Solution
Rewrite using the Quotient Property.  
Simplify the denominator.  
Rationalize the denominator.  
Simplify.  
Simplify. 
Try It 9.137
Simplify: $\sqrt{\frac{3}{27}}$.
Try It 9.138
Simplify: $\sqrt{\frac{10}{50}}$.
Rationalize a TwoTerm Denominator
When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.
When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.
Example 9.70
Simplify: $\frac{4}{4+\sqrt{2}}$.
Solution
Multiply the numerator and denominator by the conjugate of the denominator.  
Multiply the conjugates in the denominator.  
Simplify the denominator.  
Simplify the denominator.  
Remove common factors from the numerator and denominator.  
We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator. 
Try It 9.139
Simplify: $\frac{2}{2+\sqrt{3}}$.
Try It 9.140
Simplify: $\frac{5}{5+\sqrt{3}}$.
Example 9.71
Simplify: $\frac{5}{2\sqrt{3}}$.
Solution
Multiply the numerator and denominator by the conjugate of the denominator.  
Multiply the conjugates in the denominator.  
Simplify the denominator.  
Simplify the denominator.  
Simplify. 
Try It 9.141
Simplify: $\frac{3}{1\sqrt{5}}$.
Try It 9.142
Simplify: $\frac{2}{4\sqrt{6}}$.
Example 9.72
Simplify: $\frac{\sqrt{3}}{\sqrt{u}\sqrt{6}}$.
Solution
Multiply the numerator and denominator by the conjugate of the denominator.  
Multiply the conjugates in the denominator.  
Simplify the denominator. 
Try It 9.143
Simplify: $\frac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}$.
Try It 9.144
Simplify: $\frac{\sqrt{10}}{\sqrt{y}\sqrt{3}}$.
Example 9.73
Simplify: $\frac{\sqrt{x}+\sqrt{7}}{\sqrt{x}\sqrt{7}}$.
Solution
Multiply the numerator and denominator by the conjugate of the denominator.  
Multiply the conjugates in the denominator.  
Simplify the denominator.  
We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator. 
Try It 9.145
Simplify: $\frac{\sqrt{p}+\sqrt{2}}{\sqrt{p}\sqrt{2}}$.
Try It 9.146
Simplify: $\frac{\sqrt{q}\sqrt{10}}{\sqrt{q}+\sqrt{10}}$.
Media
Access this online resource for additional instruction and practice with dividing and rationalizing.
Section 9.5 Exercises
Practice Makes Perfect
Divide Square Roots
In the following exercises, simplify.
$\frac{\sqrt{50}}{10}$
$\frac{\sqrt{243}}{6}$
$\frac{3+\sqrt{27}}{9}$
$\frac{10\sqrt{200}}{20}$
$\frac{\sqrt{72}}{\sqrt{200}}$
$\frac{\sqrt{48}}{\sqrt{75}}$
ⓐ $\frac{\sqrt{10{y}^{3}}}{\sqrt{5y}}$ ⓑ $\frac{\sqrt{108{n}^{7}}}{\sqrt{243{n}^{3}}}$
$\frac{\sqrt{196{q}^{5}}}{\sqrt{484q}}$
$\frac{\sqrt{98r{s}^{10}}}{\sqrt{2{r}^{3}{s}^{4}}}$
$\frac{\sqrt{810{c}^{3}{d}^{7}}}{\sqrt{1000{c}^{5}d}}$
$\frac{\sqrt{72}}{18}$
$\frac{6\sqrt{45}}{12}$
$\frac{\sqrt{28}}{\sqrt{63}}$
$\frac{\sqrt{15{x}^{3}}}{\sqrt{3x}}$
Rationalize a OneTerm Denominator
In the following exercises, simplify and rationalize the denominator.
$\frac{8}{\sqrt{3}}$
$\frac{4}{\sqrt{5}}$
$\frac{10}{\sqrt{11}}$
$\frac{2}{5\sqrt{2}}$
$\frac{9}{2\sqrt{7}}$
$\frac{8}{3\sqrt{6}}$
$\sqrt{\frac{4}{27}}$
$\sqrt{\frac{8}{45}}$
$\sqrt{\frac{17}{192}}$
Rationalize a TwoTerm Denominator
In the following exercises, simplify by rationalizing the denominator.
ⓐ $\frac{4}{4+\sqrt{7}}$ ⓑ $\frac{7}{2\sqrt{6}}$
ⓐ $\frac{6}{6+\sqrt{5}}$ ⓑ $\frac{5}{4\sqrt{11}}$
$\frac{\sqrt{5}}{\sqrt{n}\sqrt{7}}$
$\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}$
$\frac{\sqrt{s}\sqrt{6}}{\sqrt{s}+\sqrt{6}}$
$\frac{\sqrt{80{p}^{3}q}}{\sqrt{5p{q}^{5}}}$
$\frac{3}{5\sqrt{8}}$
$\sqrt{\frac{12}{20}}$
$\frac{20}{4\sqrt{3}}$
$\frac{\sqrt{5}}{\sqrt{y}\sqrt{7}}$
$\frac{\sqrt{m}\sqrt{3}}{\sqrt{m}+\sqrt{3}}$
Everyday Math
A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify $\sqrt{\frac{250}{16}}$ to determine how many seconds it takes for the supply kit to reach the ground.
A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify $\sqrt{\frac{1200}{16}}$ to determine how many seconds it takes for the flare to reach the ocean.
Writing Exercises
 ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.
 ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.
 ⓒ Why are the two methods of simplifying square roots different?
 ⓐ Approximate $\frac{1}{\sqrt{2}}$ by dividing $\frac{1}{1.414}$ using long division without a calculator.
 ⓑ Rationalizing the denominator of $\frac{1}{\sqrt{2}}$ gives $\frac{\sqrt{2}}{2}.$ Approximate $\frac{\sqrt{2}}{2}$ by dividing $\frac{1.414}{2}$ using long division without a calculator.
 ⓒ Do you agree that rationalizing the denominator makes calculations easier? Why or why not?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are wellprepared for the next section? Why or why not?