Lesson 4

Lesson 4: Multiplying and Dividing Radicals

 

Focus

 

Skydivers jump out of an airplane and free fall through the air. As they fall, their speed increases until they open their parachutes. You can calculate a skydiver’s velocity during the first part of the skydiving jump by using the radical expression where

• d is the distance the skydiver has fallen in metres

• a is the acceleration due to gravity, which on Earth is 9.8 m/s2

• vi is the initial velocity in m/s

• vf is the final velocity in m/s at distance d

While most skydivers likely don’t calculate their velocity, it is important that the designers of parachutes have done these and other calculations so the parachute will safely return the skydivers to the ground. You don’t have to be a young physics genius to know that radical expressions are important. Knowing how to work with a radical expression and simplify it correctly is needed not just in science, but also in art, finance, human relations, political science, business, astronomy, and a host of other fields.

 

In this lesson you will multiply, divide, and simplify radical expressions. These are essential skills needed to deal effectively with the radical expressions and equations you will encounter in the next lessons.

 

This lesson will help you answer the following inquiry questions:

• How do you multiply radical expressions?
  
  
• How do you divide radical expressions?

Assessment

• Lesson 4 Assignment

All assessment items you encounter need to be placed in your course folder.

 

Save a copy of the Lesson 4 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

 

Materials and Equipment

• calculator

Launch

 

This section presents questions to help you determine if you have the skills and knowledge to complete this lesson successfully. Before beginning this lesson, you should be able to

• multiply variable expressions with exponents
  
  
• divide variable expressions with exponents

Are You Ready?

 

Answer the questions and check your solutions. If you are experiencing difficulty, you may want to use the information in the Refresher section to clarify concepts before moving on to the Discover section.

 

Simplify each of the following expressions.

1. (x3)(x2) Answers
   
   
2. (5x3)(3x4) Answers
   
   
3. Answers
   
   
4. Answers
   
   
5. Answers
   
   
6. 7x(2x2 + 3y) Answers
   
   
7. (2x + 3)2 Answers

Refresher

 

Visit Exponent to find a definition of exponents and examples.

 

 

Next, visit the Exploring Laws of Exponents—Explore It activity to review the laws for multiplying and dividing expressions involving exponents.

 

 

You can review simplifying radicals, using the Simplifying Radicals—Activity B gizmo. When you are sure you are comfortable with the skills, complete the four assessment questions at the bottom; then click on the “Check your Answers” button.

 

To watch a video to multiply binomials together, go to the Internet and type the keywords “Math Dude” into a search engine. These keywords should take you to the Montgomery County Public Schools website. On the right side of the website, look for the heading “Algebra 1 Unit 6.” Then choose the video titled “Multiplying Polynomials.”

 

Go back to the Are You Ready? section, and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

In Lesson 3 you learned that radicals cannot be added together if they have different radicands, so does not equal (Different radicands cannot be subtracted, either; therefore, does not equal ) You can, however, add two radicals together if they have the same radicand, so (For subtraction, this means )

 

Is the same true for multiplication and division? What are the restrictions and rules that apply when multiplying and dividing radicals?

 

Try This 1

 

Use a calculator to complete the Multiplying Radicals Chart.

 

Remember that you may need to include brackets in your calculator to get the correct answers. To enter , for example, you will need to include brackets around the radicands. The following keystrokes will be needed for most calculators:

 

 

 

Try This 2

 

Use a calculator to complete Dividing Radicals Chart.

 

Share 1

 

Based on your observations from both of the Try This activities, work with a partner to create a rule that applies to the multiplication and division of radical expressions.

 

Place a copy of the rule you created in your course folder.

Explore

 

In physics, as in other sciences, calculations involving radicals are often used to explain observations and predict results, even for skydiving.

 

In Lesson 3 Josh used a simple method to add or subtract radicals. As long as the radicands and indices (the plural of index) were the same, he added or subtracted the coefficients but he never changed the radicands.

 

When multiplying and dividing radicals, you need to work with both the coefficients and the radicands. For example, when multiplying radicals, the indices must be the same. If they are the same, start by multiplying the coefficients and then multiply the radicands. You cannot simplify an expression by multiplying or dividing a square root by a cube root.

 

 

In previous math courses, you learned a number of rules for multiplying and dividing monomials, binomials, and polynomials. Those same rules apply to radicals, but remember that radicals must have the same indices to be multiplied or divided.

Multiplication of Radicals

 

There are two general rules when multiplying radicals:

• 
  
  
• 

Multiplying Entire Radicals

 

Read the following or go to Multiplying Entire Radicals.

 

 

You can simplify the following expression:

 

 

 

 

Can you see an alternative solution to this problem? You could have simplified one of the original terms before multiplying.

 

Multiplying Mixed Radicals

 

Read the following or go to Multiplying Mixed Radicals.

 

 

You can simplify the following expression:

 

 

 

A slightly different approach to simplifying the same expression follows:

 

Multiplying More Complex Radical Expressions

 

Read the following or go to Multiplying Complex Radicals.

 

 

So far, all of the expressions have been monomials. The same rule applies to binomials.

 

You can use these rules to simplify the following:

 

 

 

This process can be used to simplify another binomial product. Read the following or go to Multiplying Binomial Radicals.

 

 

 

 

Self-Check 1

 

Simplify the following radicals by multiplication. For questions 2, 3, and 4, state any restrictions on the variables. Be sure to look at the original expression when stating restrictions on variables.

1. Answer
   
   
2. Answer
   
   
3. Answer
   
   
4. Answer

Dividing Radicals

 

There are two general rules when dividing radicals:

• 
  
  
• 

Dividing Entire Radicals

 

Read the following or go to Dividing Entire Radicals.

 

 

You can simplify the following expression:

 

 

Dividing Mixed Radicals

 

Read the following or go to Dividing Mixed Radicals.

 

 

You can simplify the following expression:

 

 

 

Then read the following or go to Dividing Complex Radicals.

 

 

Simplify the following complex radical:

 

 

 

Self-Check 2

 

Simplify the following radicals, and state any restrictions on the variables.

1. Answer
   
   
2. Answer
   
   
3. Answer
   
   
4. Answer

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If you need extra practice dividing radicals, work through the Simplifying Radicals—Activity A gizmo. Skip all the questions that say “Simplify by rationalizing the denominator” by clicking on the “New” button to go to the next question. When you are sure you are comfortable with the skills, complete Assessment Questions 1, 2, 4, and 5 at the bottom. Then click on the “Check Your Answers” button.

 

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Lesson 4 Summary

 

The same rules used for multiplying and dividing monomials, binomials, and polynomials in previous math courses also apply to radicals.

 

When multiplying and dividing radicals, you need to work with both the coefficients and the radicands. The key to multiplying and dividing radicals is to be sure that they have the same index. Just as you can’t mix apples and oranges, you can’t multiply or divide radicals that have different indices.

 

When multiplying radicals, first multiply the coefficients, and then multiply the radicands. When dividing radicals, first divide the coefficients, and then divide the radicands. In both cases, you must always simplify your answer so that the radical is in lowest terms.

 

You also looked at restrictions on the variables in radicals. If the index is an even number, the radicands cannot have a negative number. If the index is an odd number, the radicand could be negative or positive, so there are no restrictions on the variable. The exception to this is when you have a variable in the denominator because division by zero is undefined. If there is a variable in the denominator, then you must state the restriction that the variable cannot equal zero.

 

In the next lesson you will simplify fractional radicals further by learning to write them without a radical in the denominator.