Learn on PengiSaxon Algebra 1Chapter 11: Advanced Topics in Algebra

Lesson 103: Dividing Radical Expressions

In this Grade 9 Saxon Algebra 1 lesson, students learn to divide radical expressions using the Quotient Property of Radicals and to rationalize denominators by multiplying by a factor equivalent to 1. The lesson covers simplifying radical fractions with numeric and variable radicands, as well as using conjugates to rationalize binomial denominators containing radicals. Students practice writing radical expressions in simplest form, where the radicand contains no perfect square factors and no fractions appear under the radical symbol.

Section 1

📘 Dividing Radical Expressions

New Concept

When dividing radical expressions, use the Quotient Property of Radicals.

ab=ab, where b≠0\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } b \neq 0

What’s next

Next, you’ll apply this property and a technique called “rationalizing” to simplify various radical fractions, including those with binomial denominators.

Section 2

Quotient Property of Radicals

Property

When dividing radical expressions, use the Quotient Property of Radicals.

ab=ab, where b≠0 \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } b \neq 0

Explanation

Think of this as the 'divide and conquer' rule for square roots. Instead of tackling a fraction inside one big radical, you can split it into two separate problems: the square root of the top divided by the square root of the bottom. This makes simplifying much easier!

Examples

3625=3625=65 \sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}
1049=1049=107 \sqrt{\frac{10}{49}} = \frac{\sqrt{10}}{\sqrt{49}} = \frac{\sqrt{10}}{7}
y681=y681=y39 \sqrt{\frac{y^6}{81}} = \frac{\sqrt{y^6}}{\sqrt{81}} = \frac{y^3}{9}

Section 3

Rationalize the Denominator

Property

To rationalize a denominator means to use a method which removes radicals from the denominator of a fraction.

Explanation

It’s considered bad manners in math to leave a radical in the basement (the denominator)! To clean it up, we cleverly multiply by a form of 1, like

33 \frac{\sqrt{3}}{\sqrt{3}}
, which kicks the radical out of the denominator without changing the fraction’s value.

Examples

72=72⋅22=142 \frac{\sqrt{7}}{\sqrt{2}} = \frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{14}}{2}
5a=5a⋅aa=5aa \sqrt{\frac{5}{a}} = \frac{\sqrt{5}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{5a}}{a}

Section 4

Example Card: Simplifying Before Rationalizing

Let's tackle a complex radical fraction by simplifying first—it's easier than it looks. This example focuses on the key idea of rationalizing the denominator after simplifying.

Example Problem

Simplify 48x4227x3\frac{\sqrt{48x^4}}{2\sqrt{27x^3}}. All variables represent non-negative numbers.

Section 5

Conjugate

Property

The conjugate of an irrational number in the form

a+b a + \sqrt{b}
is
a−b a - \sqrt{b}
.

Explanation

When the denominator is a binomial team like

(2+3) (2 + \sqrt{3})
, you need its secret twin—the conjugate! Multiplying by the conjugate uses the difference of squares pattern to magically eliminate the radical from the denominator, making your expression neat and tidy.

Examples

64+2=64+2⋅4−24−2=24−6216−2=12−327 \frac{6}{4 + \sqrt{2}} = \frac{6}{4 + \sqrt{2}} \cdot \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{24 - 6\sqrt{2}}{16 - 2} = \frac{12 - 3\sqrt{2}}{7}
87−1=87−1⋅7+17+1=87+87−1=47+43 \frac{8}{\sqrt{7} - 1} = \frac{8}{\sqrt{7} - 1} \cdot \frac{\sqrt{7} + 1}{\sqrt{7} + 1} = \frac{8\sqrt{7} + 8}{7 - 1} = \frac{4\sqrt{7} + 4}{3}

Section 6

Example Card: Using Conjugates to Rationalize

A binomial in the denominator seems tricky, but its conjugate is the key to unlocking a simple answer. This example shows how to use conjugates to rationalize the denominator.

Example Problem

Simplify the expression 45+7\frac{4}{5 + \sqrt{7}}.

Book overview

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Chapter 11: Advanced Topics in Algebra

  1. Lesson 1

    Lesson 101: Solving Multi-Step Absolute-Value Inequalities

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  2. Lesson 2

    Lesson 102: Solving Quadratic Equations Using Square Roots

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  3. Lesson 3Current

    Lesson 103: Dividing Radical Expressions

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  4. Lesson 4

    Lesson 104: Solving Quadratic Equations by Completing the Square

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  5. Lesson 5

    Lesson 105: Recognizing and Extending Geometric Sequences

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  6. Lesson 6

    Lesson 106: Solving Radical Equations

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  7. Lesson 7

    Lesson 107: Graphing Absolute-Value Functions

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  8. Lesson 8

    Lesson 108: Identifying and Graphing Exponential Functions

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  9. Lesson 9

    Lesson 109: Graphing Systems of Linear Inequalities

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  10. Lesson 10

    Lesson 110: Using the Quadratic Formula

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Lesson overview

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Section 1

📘 Dividing Radical Expressions

New Concept

When dividing radical expressions, use the Quotient Property of Radicals.

ab=ab, where b≠0\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } b \neq 0

What’s next

Next, you’ll apply this property and a technique called “rationalizing” to simplify various radical fractions, including those with binomial denominators.

Section 2

Quotient Property of Radicals

Property

When dividing radical expressions, use the Quotient Property of Radicals.

ab=ab, where b≠0 \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } b \neq 0

Explanation

Think of this as the 'divide and conquer' rule for square roots. Instead of tackling a fraction inside one big radical, you can split it into two separate problems: the square root of the top divided by the square root of the bottom. This makes simplifying much easier!

Examples

3625=3625=65 \sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}
1049=1049=107 \sqrt{\frac{10}{49}} = \frac{\sqrt{10}}{\sqrt{49}} = \frac{\sqrt{10}}{7}
y681=y681=y39 \sqrt{\frac{y^6}{81}} = \frac{\sqrt{y^6}}{\sqrt{81}} = \frac{y^3}{9}

Section 3

Rationalize the Denominator

Property

To rationalize a denominator means to use a method which removes radicals from the denominator of a fraction.

Explanation

It’s considered bad manners in math to leave a radical in the basement (the denominator)! To clean it up, we cleverly multiply by a form of 1, like

33 \frac{\sqrt{3}}{\sqrt{3}}
, which kicks the radical out of the denominator without changing the fraction’s value.

Examples

72=72⋅22=142 \frac{\sqrt{7}}{\sqrt{2}} = \frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{14}}{2}
5a=5a⋅aa=5aa \sqrt{\frac{5}{a}} = \frac{\sqrt{5}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{5a}}{a}

Section 4

Example Card: Simplifying Before Rationalizing

Let's tackle a complex radical fraction by simplifying first—it's easier than it looks. This example focuses on the key idea of rationalizing the denominator after simplifying.

Example Problem

Simplify 48x4227x3\frac{\sqrt{48x^4}}{2\sqrt{27x^3}}. All variables represent non-negative numbers.

Section 5

Conjugate

Property

The conjugate of an irrational number in the form

a+b a + \sqrt{b}
is
a−b a - \sqrt{b}
.

Explanation

When the denominator is a binomial team like

(2+3) (2 + \sqrt{3})
, you need its secret twin—the conjugate! Multiplying by the conjugate uses the difference of squares pattern to magically eliminate the radical from the denominator, making your expression neat and tidy.

Examples

64+2=64+2⋅4−24−2=24−6216−2=12−327 \frac{6}{4 + \sqrt{2}} = \frac{6}{4 + \sqrt{2}} \cdot \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{24 - 6\sqrt{2}}{16 - 2} = \frac{12 - 3\sqrt{2}}{7}
87−1=87−1⋅7+17+1=87+87−1=47+43 \frac{8}{\sqrt{7} - 1} = \frac{8}{\sqrt{7} - 1} \cdot \frac{\sqrt{7} + 1}{\sqrt{7} + 1} = \frac{8\sqrt{7} + 8}{7 - 1} = \frac{4\sqrt{7} + 4}{3}

Section 6

Example Card: Using Conjugates to Rationalize

A binomial in the denominator seems tricky, but its conjugate is the key to unlocking a simple answer. This example shows how to use conjugates to rationalize the denominator.

Example Problem

Simplify the expression 45+7\frac{4}{5 + \sqrt{7}}.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 11: Advanced Topics in Algebra

  1. Lesson 1

    Lesson 101: Solving Multi-Step Absolute-Value Inequalities

    LearnPractice
  2. Lesson 2

    Lesson 102: Solving Quadratic Equations Using Square Roots

    LearnPractice
  3. Lesson 3Current

    Lesson 103: Dividing Radical Expressions

    LearnPractice
  4. Lesson 4

    Lesson 104: Solving Quadratic Equations by Completing the Square

    LearnPractice
  5. Lesson 5

    Lesson 105: Recognizing and Extending Geometric Sequences

    LearnPractice
  6. Lesson 6

    Lesson 106: Solving Radical Equations

    LearnPractice
  7. Lesson 7

    Lesson 107: Graphing Absolute-Value Functions

    LearnPractice
  8. Lesson 8

    Lesson 108: Identifying and Graphing Exponential Functions

    LearnPractice
  9. Lesson 9

    Lesson 109: Graphing Systems of Linear Inequalities

    LearnPractice
  10. Lesson 10

    Lesson 110: Using the Quadratic Formula

    LearnPractice