Lesson 3

Lesson 3: Rationalizing Radical Expressions

Focus

diver on cliff: Photodisc/Thinkstock; diver in air: Brand X Pictures/Thinkstock

Cliff divers at Acapulco, Mexico, leap out from the cliff face and plunge 35 m into a shallow gorge below. That is comparable to diving off a 12-storey building! In the gorge, the water depth varies from 2 m to 5 m, depending on the wave action. Timing a jump is essential.

A formula to calculate the time it takes for the diver to hit the water is . This radical expression is derived from formulas learned in high school physics where t represents time, d represents distance, and a represents the acceleration due to gravity.

In this lesson you will learn to simplify fractional radicals like the one in this example by writing them without a radical in the denominator.

Outcomes

At the end of this lesson you will be able to

• simplify expressions with radical denominators by changing the denominator to a natural number for both monomial and binomial denominators
  
  
• describe the relationship between rationalizing a binomial denominator of a rational expression and the product of the factors of a difference-of-squares expression
  
  
• explain, using examples, that the square root of a number squared refers to the absolute value of that number

Lesson Questions

You will investigate the following questions:

• How do you simplify a fractional radical by rationalizing the denominator?
  
  
• How do you explain the rules for writing positive and negative roots by using examples?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 5 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

Before beginning this lesson you should be able to

• define commutative property
  
  
• define natural numbers
  
  
• use the distributive property when multiplying binomials
  
  
• recognize the factors for a difference-of-squares binomial

Are You Ready?

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Define commutative property. Answer
   
   
2. Define natural numbers. Answer
   
   
3. Expand the following expressions by using the distributive property to eliminate the brackets.
   
   
   1. 5(x3 + 2) Answer
      
      
   2. 4(x2 − 12) Answer
      
      
   3. 2x(x3 + x2) Answer
      
      
   4. 5xy(x3 − 3y4) Answer
      
      
4. Use the distributive property when multiplying these binomials.
   
   
   1. (x − 1)(x − 3) Answer
      
      
   2. (5x − 3)(3x − 2) Answer
      
      
   3. (2x + 1)(2x − 1) Answer
      
      
   4. (5x − 3)(5x + 3) Answer
      
      
5. In which parts of question 4 are you multiplying the factors for the difference of squares? Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

Review Commutative Property.

Review the definition, study examples, and try an applet in Natural Numbers.

Visit Distributive Property for a definition and examples to keep in mind when multiplying binomials.

Watch “Factoring Difference of Squares.”

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

Discover

In this section you will investigate the validity of the following statements:

Try This 1

1. Use a calculator to see whether the square root of a number multiplied by the square root of another number is equal to the square root of the product of the numbers. In simple terms, is true?
   
   
   • Try some of the common numbers provided in the chart until you think you know the answer.
     
     
   • Enter some of your own numbers to check.
     
     
   • If you prefer, you may use a spreadsheet to do these calculations.
     
   
   
    

   a	b	(to two decimal places)	(to two decimal places)
   1	10
   2	9
   3	8
   4	7
   5	6
   6	4
   7	3
   8	5
   9	4

   
   
   
2. As you were completing the columns for and , did you notice any patterns? If so, explain?
   
   
3. Does ? Explain your reasoning.
   
   
4. Use a calculator to determine whether . In words, is the square root of some number divided by the square root of another number equal to the square root of the quotient of the two numbers? Try some of the common numbers in the chart until you think you know the answer. Enter some of your own numbers to check.
   
   
    

   a	b	(to two decimal places)	(to two decimal places)
   10	2
   9	3
   8	4
   7	5
   6	6
   4	7
   3	8
   5	9
   4	10

   
   
   
5. What do you conclude? Is ?

Save your work in your course folder.

Share 1

1. Based on your observations from Try This 1, discuss the following questions with a partner or group.
   
   
   1. Is ?
      
      
   2. Is ?
      
      
2. Summarize your discussion by creating a general rule about multiplying square roots and another rule about dividing square roots.

Save your work in your course folder.

Explore

Ryan McVay/Digital Vision/Thinkstock

Strategies for when the Radical is in the Denominator

In Discover, you may have found that the following statements are true:

Introducing Conjugates

The binomial factors for the difference of squares are called conjugates. In the equation , the conjugates are (x − y) and (x + y). Conjugates may contain radicals; for example, .

Example

If you had difficulty solving in Try This 3, this example should help. You may want to revisit Try This 3 after reading this example.

This example shows how to rationalize an expression with a binomial denominator containing a radical. First, determine the conjugate of the denominator. Then multiply both the numerator and the denominator by this conjugate. This will eliminate the radical from the denominator.

Consider .

The binomial denominator is .

The conjugate of . Multiply the numerator and denominator by the conjugate .

Notice that by multiplying the expression by the conjugate, the radical in the numerator is simplified to an integer. This will be the case every time.

Simplify:

More Examples

“How to Rationalize the Denominator” summarizes the different strategies you have now learned to eliminate the radical from the denominator. The video progresses from basic examples to more complex problems.

Self-Check 2

1. Simplify the following radicals by rationalizing the denominator. Verify your answers using a calculator.
   
   
   1. Answer
      
      
   2. Answer
      
      
   3. Answer
      
      

2. The highest diving platforms at pools are 10 m above the water. The time a diver takes to hit the water can be calculated using the radical equation . Assume the distance, d , is –10 m. The acceleration due to gravity, a, is –9.81 m/s2.

   
   
   © Hristo Momcharov/10939305/Fotolia
   The negative sign in both the values given indicates that the direction is downward, since the diver is falling. Since you cannot take the square root of negative numbers, rationalize the denominator before substituting the values in for the variables. Use the resulting expression and the data given to find the time for a dive to the nearest tenth of a second. Answer
   

Try This 4

When you square a negative number, the result is positive; for example, (−5)2 = 25. However, the square-root sign indicates only the absolute value of the square root; , not −5.

How do you indicate that you want the negative root? How do you indicate that you want both the negative and the positive root?

1. Use symbols to indicate that you want only the negative root of x2.
   
   
2. Use symbols to indicate that you want both the negative and positive roots of y4.
   
   

Self-Check 3

Practise your new skills in Positive and Negative Roots.

Skip forward to Connect if you feel you have a solid understanding of how to

• simplify expressions with radical denominators by rationalizing the denominator
  
  
• explain the rules for writing positive and negative roots, including the idea that the square root of a number squared refers to the absolute value of that number

If you need a bit more practice, complete Self-Check 4.

Self-Check 4

Connect

Lesson 3 Assignment

Lesson 3 Summary

In this lesson you investigated the following questions:

• How do you simplify a fractional radical by rationalizing the denominator?

• How do you explain the rules for writing positive and negative roots by using examples?

Watch Module 5: Lesson 3 Summary.