Lynn Marecek; Andrea Honeycutt Mathis

Learning Objectives

By the end of this section, you will be able to:

Divide radical expressions
Rationalize a one term denominator
Rationalize a two term denominator

Be Prepared 8.13

Before you get started, take this readiness quiz.

Simplify: $\frac{30}{48} .$
If you missed this problem, review Example 1.24.

Be Prepared 8.14

Simplify: $x^{2} \cdot x^{4} .$
If you missed this problem, review Example 5.12.

Be Prepared 8.15

Multiply: $(7 + 3 x) (7 - 3 x) .$
If you missed this problem, review Example 5.32.

Divide Radical Expressions

We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals.

We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars are needed.

Quotient Property of Radical Expressions

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b \neq 0,$ and for any integer $n \geq 2$ then,

\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}

We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.

Example 8.47

Simplify: ⓐ $\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$ ⓑ $\frac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}} .$

Solution

ⓐ

	$\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$
Rewrite using the quotient property, $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} .$	$\sqrt{\frac{72 x^{3}}{162 x}}$
Remove common factors.	$\sqrt{\frac{18 \cdot 4 \cdot x^{2} \cdot x}{18 \cdot 9 \cdot x}}$
Simplify.	$\sqrt{\frac{4 x^{2}}{9}}$
Simplify the radical.	$\frac{2 x}{3}$

ⓑ

	$\frac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}}$
Rewrite using the quotient property, $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} .$	$\sqrt[3]{\frac{32 x^{2}}{4 x^{5}}}$
Simplify the fraction under the radical.	$\sqrt[3]{\frac{8}{x^{3}}}$
Simplify the radical.	$\frac{2}{x}$

Try It 8.93

Simplify: ⓐ $\frac{\sqrt{50 s^{3}}}{\sqrt{128 s}}$ ⓑ $\frac{\sqrt[3]{56 a^{}}}{\sqrt[3]{7 a^{4}}} .$

Try It 8.94

Simplify: ⓐ $\frac{\sqrt{75 q^{5}}}{\sqrt{108 q}}$ ⓑ $\frac{\sqrt[3]{72 b^{2}}}{\sqrt[3]{9 b^{5}}} .$

Example 8.48

Simplify: ⓐ $\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$ ⓑ $\frac{\sqrt[3]{−250 m^{} n^{−2}}}{\sqrt[3]{2 m^{−2} n^{4}}} .$

Solution

ⓐ

	$\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$
Rewrite using the quotient property.	$\sqrt{\frac{147 a b^{8}}{3 a^{3} b^{4}}}$
Remove common factors in the fraction.	$\sqrt{\frac{49 b^{4}}{a^{2}}}$
Simplify the radical.	$\frac{7 b^{2}}{a}$

ⓑ

	$\frac{\sqrt[3]{−250 m^{} n^{−2}}}{\sqrt[3]{2 m^{−2} n^{4}}}$
Rewrite using the quotient property.	$\sqrt[3]{\frac{−250 m^{} n^{−2}}{2 m^{−2} n^{4}}}$
Simplify the fraction under the radical.	$\sqrt[3]{\frac{−125 m^{3}}{n^{6}}}$
Simplify the radical.	$- \frac{5 m}{n^{2}}$

Try It 8.95

Simplify: ⓐ $\frac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}$ ⓑ $\frac{\sqrt[3]{−128 x^{2} y^{−1}}}{\sqrt[3]{2 x^{−1} y^{2}}} .$

Try It 8.96

Simplify: ⓐ $\frac{\sqrt{300 m^{3} n^{7}}}{\sqrt{3 m^{5} n}}$ ⓑ $\frac{\sqrt[3]{−81 p q^{−1}}}{\sqrt[3]{3 p^{−2} q^{5}}} .$

Example 8.49

Simplify: $\frac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}} .$

Solution

	$\frac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}$
Rewrite using the quotient property.	$\sqrt{\frac{54 x^{5} y^{3}}{3 x^{2} y}}$
Remove common factors in the fraction.	$\sqrt{18 x^{3} y^{2}}$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{9 x^{2} y^{2} \cdot 2 x}$
Rewrite the radical as the product of two radicals.	$\sqrt{9 x^{2} y^{2}} \cdot \sqrt{2 x}$
Simplify.	$3 x y \sqrt{2 x}$

Try It 8.97

Simplify: $\frac{\sqrt{64 x^{4} y^{5}}}{\sqrt{2 x y^{3}}} .$

Try It 8.98

Simplify: $\frac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}} .$

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator 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. 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.

This process is still used today, and is useful in other areas of mathematics, too.

Rationalizing the Denominator

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

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 radical.

Similarly, a radical expression is not considered simplified if the radicand contains a fraction.

Simplified Radical Expressions

A radical expression is considered simplified if there are

no factors in the radicand have perfect powers of the index
no fractions in the radicand
no radicals in the denominator of a fraction

To rationalize a denominator with a square root, we use the property that ${(\sqrt{a})}^{2} = a .$ 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 8.50

Simplify: ⓐ $\frac{4}{\sqrt{3}}$ ⓑ $\sqrt{\frac{3}{20}}$ ⓒ $\frac{3}{\sqrt{6 x}} .$

Solution

To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ


Multiply both the numerator and denominator by $\sqrt{3} .$
Simplify.

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect square, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt{5} .$
Simplify.
Simplify.

ⓒ


Multiply the numerator and denominator by $\sqrt{6 x} .$
Simplify.
Simplify.

Try It 8.99

Simplify: ⓐ $\frac{5}{\sqrt{3}}$ ⓑ $\sqrt{\frac{3}{32}}$ ⓒ $\frac{2}{\sqrt{2 x}} .$

Try It 8.100

Simplify: ⓐ $\frac{6}{\sqrt{5}}$ ⓑ $\sqrt{\frac{7}{18}}$ ⓒ $\frac{5}{\sqrt{5 x}} .$

When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.

We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.

For example,

Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.

We will use this technique in the next examples.

Example 8.51

Simplify ⓐ $\frac{1}{\sqrt[3]{6}}$ ⓑ $\sqrt[3]{\frac{7}{24}}$ ⓒ $\frac{3}{\sqrt[3]{4 x}} .$

Solution

To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ


The radical in the denominator has one factor of 6. Multiply both the numerator and denominator by $\sqrt[3]{6^{2}},$ which gives us 2 more factors of 6.
Multiply. Notice the radicand in the denominator has 3 powers of 6.
Simplify the cube root in the denominator.

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect cube, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt[3]{3^{2}} .$ This will give us 3 factors of 3.
Simplify.
Remember, $\sqrt[3]{3^{3}} = 3 .$
Simplify.

ⓒ


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by $\sqrt[3]{2 \cdot x^{2}} .$ This will get us 3 factors of 2 and 3 factors of x.
Simplify.
Simplify the radical in the denominator.

Try It 8.101

Simplify: ⓐ $\frac{1}{\sqrt[3]{7}}$ ⓑ $\sqrt[3]{\frac{5}{12}}$ ⓒ $\frac{5}{\sqrt[3]{9 y}} .$

Try It 8.102

Simplify: ⓐ $\frac{1}{\sqrt[3]{2}}$ ⓑ $\sqrt[3]{\frac{3}{20}}$ ⓒ $\frac{2}{\sqrt[3]{25 n}} .$

Example 8.52

Simplify: ⓐ $\frac{1}{\sqrt[4]{2}}$ ⓑ $\sqrt[4]{\frac{5}{64}}$ ⓒ $\frac{2}{\sqrt[4]{8 x}} .$

Solution

To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ


The radical in the denominator has one factor of 2. Multiply both the numerator and denominator by $\sqrt[4]{2^{3}},$ which gives us 3 more factors of 2.
Multiply. Notice the radicand in the denominator has 4 powers of 2.
Simplify the fourth root in the denominator.

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect fourth power, so rewrite using the Quotient Property.
Rewrite the radicand in the denominator to show the factors.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt[4]{2^{2}} .$ This will give us 4 factors of 2.
Simplify.
Remember, $\sqrt[4]{2^{4}} = 2 .$
Simplify.

ⓒ


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by $\sqrt[4]{2 \cdot x^{3}} .$ This will get us 4 factors of 2 and 4 factors of x.
Simplify.
Simplify the radical in the denominator.
Simplify the fraction.

Try It 8.103

Simplify: ⓐ $\frac{1}{\sqrt[4]{3}}$ ⓑ $\sqrt[4]{\frac{3}{64}}$ ⓒ $\frac{3}{\sqrt[4]{125 x}} .$

Try It 8.104

Simplify: ⓐ $\frac{1}{\sqrt[4]{5}}$ ⓑ $\sqrt[4]{\frac{7}{128}}$ ⓒ $\frac{4}{\sqrt[4]{4 x}}$

Rationalize a Two Term 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.

\begin{matrix} (a - b) (a + b) & (2 - \sqrt{5}) (2 + \sqrt{5}) \\ a^{2} - b^{2} & 2^{2} - {(\sqrt{5})}^{2} \\ 4 - 5 \\ −1 \end{matrix}

When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

Example 8.53

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 8.105

Simplify: $\frac{3}{1 - \sqrt{5}} .$

Try It 8.106

Simplify: $\frac{2}{4 - \sqrt{6}} .$

Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.

Example 8.54

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 8.107

Simplify: $\frac{\sqrt{5}}{\sqrt{x} + \sqrt{2}} .$

Try It 8.108

Simplify: $\frac{\sqrt{10}}{\sqrt{y} - \sqrt{3}} .$

Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.

Example 8.55

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. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.

Try It 8.109

Simplify: $\frac{\sqrt{p} + \sqrt{2}}{\sqrt{p} - \sqrt{2}} .$

Try It 8.110

Simplify: $\frac{\sqrt{q} - \sqrt{10}}{\sqrt{q} + \sqrt{10}}$

Media

Access these online resources for additional instruction and practice with dividing radical expressions.

Section 8.5 Exercises

Practice Makes Perfect

Divide Square Roots

In the following exercises, simplify.

245.

ⓐ $\frac{\sqrt{128}}{\sqrt{72}}$ ⓑ $\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

246.

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$ ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

247.

ⓐ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$ ⓑ $\frac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}$

248.

ⓐ $\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}$ ⓑ $\frac{\sqrt[3]{54 y^{}}}{\sqrt[3]{16 y^{4}}}$

249.

ⓐ $\frac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}$ ⓑ $\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}$

250.

ⓐ $\frac{\sqrt{196 q^{}}}{\sqrt{484 q^{5}}}$ ⓑ $\frac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m^{}}}$

251.

ⓐ $\frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}$ ⓑ $\frac{\sqrt[3]{−16 a^{4} b^{−2}}}{\sqrt[3]{2 a^{−2} b^{}}}$

252.

ⓐ $\frac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}$ ⓑ $\frac{\sqrt[3]{−375 y^{4} z^{−2}}}{\sqrt[3]{3 y^{−2} z^{4}}}$

253.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$ ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

254.

ⓐ $\frac{\sqrt{810 c^{−3} d^{7}}}{\sqrt{1000 c^{} d^{−1}}}$ ⓑ $\frac{\sqrt[3]{24 a^{7} b^{- 1}}}{\sqrt[3]{- 81 a^{- 2} b^{2}}}$

255.

$\frac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}$

256.

$\frac{\sqrt{72 a^{3} b^{6}}}{\sqrt{3 a b^{3}}}$

257.

$\frac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{- 1} b^{3}}}$

258.

$\frac{\sqrt[3]{162 x^{- 3} y^{6}}}{\sqrt[3]{2 x^{3} y^{- 2}}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

259.

260.

261.

262.

263.

264.

265.

266.

267.

268.

269.

270.

Rationalize a Two Term Denominator

In the following exercises, simplify.

271.

$\frac{8}{1 - \sqrt{5}}$

272.

$\frac{7}{2 - \sqrt{6}}$

273.

$\frac{6}{3 - \sqrt{7}}$

274.

$\frac{5}{4 - \sqrt{11}}$

275.

$\frac{\sqrt{3}}{\sqrt{m} - \sqrt{5}}$

276.

$\frac{\sqrt{5}}{\sqrt{n} - \sqrt{7}}$

277.

$\frac{\sqrt{2}}{\sqrt{x} - \sqrt{6}}$

278.

$\frac{\sqrt{7}}{\sqrt{y} + \sqrt{3}}$

279.

$\frac{\sqrt{r} + \sqrt{5}}{\sqrt{r} - \sqrt{5}}$

280.

$\frac{\sqrt{s} - \sqrt{6}}{\sqrt{s} + \sqrt{6}}$

281.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

282.

$\frac{\sqrt{m} - \sqrt{3}}{\sqrt{m} + \sqrt{3}}$

Writing Exercises

283.

ⓐ 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?

284.

Explain what is meant by the word rationalize in the phrase, “rationalize a denominator.”

285.

Explain why multiplying $\sqrt{2 x} - 3$ by its conjugate results in an expression with no radicals.

286.

Explain why multiplying $\frac{7}{\sqrt[3]{x}}$ by $\frac{\sqrt[3]{x}}{\sqrt[3]{x}}$ does not rationalize the denominator.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “divide radical expressions.”, “rationalize a one term denominator”, and “rationalize a two term denominator”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

8.5 Divide Radical Expressions

Divide Radical Expressions

Solution

Solution

Solution

Rationalize a One Term Denominator

Solution

Solution

Solution

Rationalize a Two Term Denominator

Solution

Solution

Solution

Section 8.5 Exercises

Practice Makes Perfect

Writing Exercises

Self Check