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Lesson 2

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Table of contents

1. Lesson 2

Mathematics 20-1 Module 5

Module 5: Radicals

 

Lesson 2: Operations on Radicals

 
Focus

 

This is a photo of two students observing a solution in a beaker.

© Sandor Kacso/12225139/Fotolia

The rate of a chemical reaction is extremely important. The rate may determine whether a chemical is violently explosive, very useful, or whether it will remain as a pollutant in the environment for decades. When substances A and B combine to produce C, a radical equation involving the rate of reaction and concentration of substances is .

 

Knowing how to work with and correctly simplify a radical expression is needed in all branches of science, not just chemistry and astronomy. In this lesson you will enhance your ability to work with radicals, including adding, subtracting, multiplying, dividing, and simplifying radicals and their components. These radical equations will model situations on this planet and throughout the universe.

 

Outcomes

 

At the end of this lesson you will be able to

  • simplify radical expressions with numerical or variable radicands by performing one or more mathematical operations
  • identify the values of the variable for which a given radical expression is defined
Lesson Questions

 

You will investigate the following questions:

  • How do you simplify a complex radical by correctly performing addition, subtraction, multiplication, and/or division?

  • For what values of the variable is a given radical expression defined?
Assessment

 

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

  • completion of the Lesson 2 Assignment (Download the Lesson 2 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


1.1. Launch

Mathematics 20-1 Module 5

Module 5: Radicals

 

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

  • add and subtract variables with exponents

  • multiply and divide variables with exponents


1.2. Are You Ready?

Mathematics 20-1 Module 5

Module 5: Radicals

 

Are You Ready?

 

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

  1. Is it possible to add or subtract variables in these forms? If it is, provide the answer. If not, explain why.

    1. 5x3 + 3x3 Answer

    2. 4x3 − 2x4 Answer

    3. x3 + x2 Answer

    4. 5x3 − 3x3 Answer

    5. 3(x − 2)2 + 4(x − 2)2 Answer

  2. Simplify the following expressions:

    1. (x3)(x2) Answer

    2. (5x3)(3x4) Answer

    3. Answer

    4. Answer

    5. 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.


 

1.3. Refresher

Mathematics 20-1 Module 5

Module 5: Radicals

 

Refresher

 

This is a play button that opens Like Terms.

Read Like Terms to prepare for adding and subtracting variables with exponents.


Watch the following two videos about multiplying and dividing variables with exponents:

  • “Exponent Properties Involving Products,” which describes many examples of multiplying with exponents

     
    This is a play button that opens “Exponent Properties Involving Products.”
    Khan Academy CC BY-NC-SA 3.0


  • “Exponent Properties Involving Quotients,” which describes many examples of dividing with exponents

     
    This is a play button that opens “Exponent Properties Involving Quotients.”
    Khan Academy CC BY-NC-SA 3.0


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



1.4. Discover

Mathematics 20-1 Module 5

Module 5: Radicals

 

Discover

 

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



Try This 1
  1. Use a calculator to determine if the square root of a number plus the square root of another number is equal to the square root of the sum of the numbers. In other words, is ? Try some of the common numbers in the chart, and then try some numbers of your own.

    If you prefer, you may use a spreadsheet to do these calculations. hint

     
    a b

    (to two decimal places)

    (to two decimal places)
    1 2  
    2 2    
    3 2    
    4 2    
    5 2    
    3 3    
    4 3    
    5 3    
    6 3    
           
           
           
           


  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 verify if the following simplification is true:

     


    Try some of the common numbers in the chart, and then try some of your own numbers.

     
    a b x

    (to two decimal places)

    (to two decimal places)
    1 2 3    
    2 2 3    
    3 2 3    
    4 2 3    
    5 2 3    
    3 3 5    
    4 3 5    
    5 3 5    
    6 3 5    
             
             
             
             


  5. What do you conclude? Is ?

course folder Save your work in your course folder.

 

Share 1
  1. Based on your observations thus far, discuss the following questions with a partner or group.

    1. Is ?

    2. Is ?

  2. Summarize your discussion by creating a general rule about adding square roots.

course folder Save your work in your course folder.


Instructions for using a spreadsheet to take roots can be found by searching the Internet for “taking roots spreadsheet.” If you put the numbers into a spreadsheet in columns A and B, beginning in row 2, column C should be =SQRT(A2) +SQRT(B2), and column D should be =SQRT(A2+B2).

 

To avoid having to input the directions into every cell individually, click on cell C2 and drag the directions down 10 cells. Then hold down the Ctrl key and press D. You can use the Ctrl + D feature to copy the instructions down each column as far as you wish.


1.5. Explore

Mathematics 20-1 Module 5

Module 5: Radicals

 

Explore

 

This is a photo of a woman standing in front of a blackboard. On the blackboard chemical equations, including radicals, are written.

iStockphoto/Thinkstock

In chemistry, as in other sciences, calculations involving radicals are often used to explain observations and predict results. In Discover, you found that you could add radicals if the radicals were like terms. The same is true of subtracting radicals.

 

Recall that radicals are like terms when the radicands and indexes are identical.

 

Example

 

are like terms, since each has a radicand of 6 and an index of 2 (since the square root is taken for both).

 

Non-Example

 

are NOT like terms, since has an index of 3 and has an index of 2. Remember that ; the index is understood!


glossary

You already saved Module 5 Glossary Terms in your course folder. You will want to define the following term, and possibly others, in your copy of Module 5 Glossary Terms:

  • like terms


1.6. Explore 2

Mathematics 20-1 Module 5

Module 5: Radicals

 

Self-Check 1
  1. Consider the mixed radical . Which of the following could be added to or subtracted from ? What will be the sum or difference of the like terms?

    1. Answer

    2. Answer

    3. Answer

    4. Answer

  2. Simplify the following radicals and combine like terms.

    1. hint Answer

    2. hint Answer

  3. Simplify the following radicals and combine like terms.

    1. Answer

    2. Answer

    3. Answer

    4. Answer
Try This 2

 

When you multiply the radical by , will the answer be 6 × 3 or ?

  1. Predict what the answer will be.

  2. Design a way to test this and record your observations.

  3. Based on your test, develop a rule to describe this situation.

  4. Test your rule using different values.

course folder Save your work in your course folder.

 

Share 2

 

Share the rule you developed in Try This 2 with a partner or group. How do your rules compare?

 

course folder Save your work in your course folder.

Look for perfect square factors of the radicand that can be simplified.
Look for perfect square factors of the radicand that can be simplified.

 

1.7. Explore 3

Mathematics 20-1 Module 5

Module 5: Radicals

 

Multiplying Mixed Radicals with Identical Radicands


textbook

Turn to “Example 1,” part b., on page 284 of the textbook to see how you can multiply mixed radicals with identical radicands.

 

Self-Check 2

 

Apply the rule you developed in Try This 2 to the following question.

  1. Simplify the following radicals by multiplying. Verify the answer to 1.a. using a calculator and computing the numerical value of both the question and the answer. For what values of the variables in 1.b., 1.c., and 1.d. do the radicals represent real numbers?

    1. Answer

    2. Answer

    3. Answer

    4. Answer
Try This 3

 

When you divide the radical , will the answer be 8 or ?

  1. Predict what the answer will be.

  2. Design a way to test this and record your observations.

  3. Based on your test, develop a rule to describe this situation.

  4. Test your rule using different values.

course folder Save your work in your course folder.

 

Share 3

 

Share the rule you developed in Try This 3 with a partner or group. How do your rules compare?

 

course folder Save your work in your course folder.

 

Self-Check 3
  1. Simplify the following radicals. Check the answer to 1.a. and 1.b. using a calculator and computing the numerical value of both the question and the answer.

    1. Answer

    2. Answer

    3. Answer

    4. Answer

  2. For what values of the variables in 1.c. and 1.d. do the radicals represent real numbers? Answer

1.8. Explore 4

Mathematics 20-1 Module 5

Module 5: Radicals

 

Simplifying Radicals by Multiplying and Dividing

 

In Try This 2 and 3 you developed and tested the following rules:

 

Radicals can be simplified by multiplying and dividing if they have the same index: 

  • Multiply the coefficients and multiply the radicands, .

  • Divide the coefficients and divide the radicands .
In the case of division, n must be a natural number, b and y cannot equal zero, and a, b, x, and y must be real numbers. If n is even, x ≥ 0 and y > 0.


formula

Add these rules to your copy of Formula Sheet.

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

  • simplify a complex radical by correctly performing addition, subtraction, multiplication, and/or division

  • identify the values of the variable for which a given radical expression is defined

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

 

Self-Check 4


textbook
Complete any or all of questions 9, 10, 12, and 14 from “Practice” on pages 279 and 280, as well as questions 1, 6.a., 6.b., 6.c., and 16 on pages 289 to 291 of the textbook. As you finish each part of a question, check your work against the answers given at the back of the textbook. If you are still unclear about how to answer some questions, ask your teacher about those questions and get some help.

 

1.9. Connect

Mathematics 20-1 Module 5

Module 5: Radicals

 

Connect

 

Lesson 2 Assignment


assignment

Open your copy of Lesson 2 Assignment, which you saved in your course folder at the beginning of this lesson. Complete the assignment.

 

course folder Save your work in your course folder.

 

Project Connection


assignment

Begin working on Activity 1 of Module 5 Project: Plan a Planet.

 

course folder Save your work in your course folder.

 

Going Beyond

 

This photo shows a young girl smelling a flower.

Jupiterimages/Creatas/Thinkstock

 

One of the laws of chemistry that uses radicals is Graham’s Law of Effusion. The law allows you to compare how fast different gases will spread out based on their molecular mass. Graham’s Law of Effusion is the basis for how soon you will smell a perfume or enjoy a reed room freshener.


 

isearch

Find out more about how the law works and its applications by searching online for “Graham’s Law.”


 

1.10. Lesson 2 Summary

Mathematics 20-1 Module 5

Module 5: Radicals

 

Lesson 2 Summary

 

This is an image of a young man looking at a failed science experiment.

Getty Images/Hemera

In this lesson you investigated the following questions:

  • How do you simplify a complex radical by correctly performing addition, subtraction, multiplication, and/or division?
  • For what values of the variable is a given radical expression defined?

You refreshed your knowledge about how to add, subtract, multiply, and divide radicals. You learned that radicals can only be added or subtracted if they are like terms.

 

You also learned that radicals have to have the same index to be multiplied. When multiplying radicals, you multiply the coefficients, and then you multiply the radicands. If the index is an even number, the radicands must not have a negative value.

 

You found that radicals have to have the same index to be divided. When dividing radicals, you first divide the coefficients, and then you divide the radicands. Divisors cannot have a value of zero. If the index is an even number, the radicands must not be negative.

 

You used this knowledge to simplify various radical expressions, including removing roots from under the radical sign where possible.

 

In the next lesson you will simplify fractional radicals further by learning to write these radicals without a radical in the denominator. You will apply the commutative principle to radicals.