Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jennifer Beddoe*

When dividing radical expressions, we use the quotient rule. This lesson will describe the quotient rule and how to use it to solve these radical expressions.

A **quotient** is the answer to a division problem. When dividing radical expressions, we use the **quotient rule** to help solve them. The quotient rule states that a radical involving two quotients is equal to the quotient of two radicals. Written out in math terms, this means:

So when you divide one radical expression by another, you can simplify it by writing both expressions under the same radical, then simplifying. The quotient rule works only if:

1. Each radical has the same **index**. The index is the superscript number to the left of the radical symbol, which indicates the degree of the radical.

In this example, the index is the 3 and it is indicating the cube root of 27. If there is no index, it is understood to be 2, or a square root.

2. The denominator of the fraction is not zero.

Here are the steps to dividing radical expressions.

- Ensure that the index of each radical is the same and that the denominator is not zero.
- Convert the expression to one radical.
- Simplify where possible.
- Rationalize the denominator, if necessary.

For example, solve:

Once we have determined that both the numerator and denominator have the same index (2) and that the denominator is not zero, we can use the quotient rule to convert the expression to this:

Solving under the radical - 100/4 = 25 - gives us âˆš25, which is equal to 5. Since there is no denominator to rationalize, the problem is finished.

Mathematics has its own language, and like any other foreign language, there are certain rules that must be followed to speak the language correctly. Speaking math correctly ensures that mathematicians all around the world are able to understand each other.

One rule for radicals is that for a radical expression to be in its simplest form, it cannot have a radical in the denominator. So the last step in simplifying any radical expression with a fraction is to make sure that there is not a radical in the denominator. This is called rationalizing the denominator.

To rationalize a denominator, you multiply the expression by an appropriate fraction that is equivalent to one. For example, simplify:

Because there is a radical in the denominator of this expression, we need to rationalize the denominator to simplify the expression so it is written mathematically correct. The first step is to create a fraction that is equivalent to one that will help remove the radical from the denominator. Most of the time, the fraction needed will be the same as the radical in the denominator. To make it a fraction that is equivalent to one, you must have the same numerator as denominator. For this example, the first step of the problem will look like this:

The simplification of this problem looks like this:

Because the square root of x^2 is equal to x, the radical is removed from the denominator and the simplification of this expression is

In order to divide more complex radical expressions, we must not only divide but make sure that there is not a radical in the denominator. Try this example. Simplify:

The first step is to determine if there are any terms that can be simplified by division before we worry about the radical. There are two terms that can be simplified in this expression: 12/6 = 2, and the *b* term in the numerator cancels the *b* term in the denominator. That leaves us with this term to simplify.

Since we can't leave the expression with a radical in the denominator, we need to rationalize it, or find something that when multiplied by the denominator will remove the radical.

When we multiply both the numerator and the denominator by the square root of *c* we get our final answer.

Here is another example to try.

The first step is to see if there are any terms we can simplify by dividing. Since 12 is not divisible by 5, and there are no variables common in the numerator and denominator, we can't do any simplifying now. So we need to break apart the numerator and denominator per the quotient rule in order to move on.

Since there is a radical in the denominator of this expression, the next step is to rationalize the denominator. What can be multiplied to the denominator so that the result will not involve a radical? The answer is:

because

which simplifies into

Next, we need to multiply the numerator and denominator by the term that will remove the radical from the denominator.

This simplifies to:

To divide radical expressions, you first must determine if the numerator and denominator can be simplified by division. After any simplifying, you need to make sure that there is no radical in the denominator. If there is, you need to simplify it by rationalizing the denominator. This is done by determining a term that when multiplied to the denominator will cancel out the radical. Then, that term is multiplied to both the numerator and denominator, everything is simplified and the resulting term is the answer.

Once you are done with this lesson, you should be able to divide a radical expression by simplifying or rationalizing, multiplying, and simplifying for the final answer.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
11 in chapter 7 of the course:

Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Division and Reciprocals of Radical Expressions 5:53
- Radicands and Radical Expressions 4:29
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Rationalizing Denominators in Radical Expressions 7:01
- Addition and Subtraction Using Radical Notation 3:08
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Go to High School Algebra: Radical Expressions

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Early Civilizations & The Ancient Near East
- Fundamental Overview of World War I
- The Virginia Dynasty & Jacksonian America
- 1920's America and the Great Depression
- Building the United States After the American Revolution
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Subtraction in Java: Method, Code & Examples
- Hydrogen Chloride vs. Hydrochloric Acid
- Extraction of Aluminum, Copper, Zinc & Iron
- Iroquois Culture, Traditions & Facts
- Noun Clauses Lesson Plan
- Adverb of Manner Lesson Plan
- Timeline Project Ideas for High School
- Quiz & Worksheet - Multi-Dimensional Arrays in C
- Quiz & Worksheet - What is a Diastereoisomer?
- Quiz & Worksheet - Mauryan Empire Art & Culture
- Quiz & Worksheet - What is a Convergent Sequence?
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Common Core English & Reading Worksheets & Printables
- Calculus Worksheets

- Praxis PLT - Grades 7-12 (5624): Practice & Study Guide
- Marketing for Real Estate Agents
- High School Geometry: Tutoring Solution
- FTCE Elementary Education K-6 (060): Practice & Study Guide
- User Experience Design Training
- TExMaT Master Reading Teacher: Comprehension Strategies
- MTTC History: Federalism in the U.S.
- Quiz & Worksheet - Equating Complex Numbers
- Quiz & Worksheet - Alliteration in Annabel Lee
- Quiz & Worksheet - Characteristics of Cubism
- Quiz & Worksheet - First Fig By Edna St. Vincent Millay
- Quiz & Worksheet - Pacing Yourself When Speaking

- Measuring Supply Chain Performance: Key Performance Indicators
- Addition Reactions of Alkenes
- Using an Online Tutor
- Common Core State Standards in Maryland
- Constellations for Kids: Projects & Activities
- How to Pass Algebra 1
- How to Pass Multiple Choice Tests
- 6th Grade Summer Reading List
- Number Games for Kids
- Civil Rights Activities for Kids
- Common Core State Standards in Maryland
- 6th Grade Writing Prompts

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject