Lesson 1
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| Course: | Math 20-1 SS |
| Book: | Lesson 1 |
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| Date: | Monday, 15 September 2025, 2:29 PM |
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1. Lesson 1
Module 5: Radicals
Lesson 1: Mixed and Entire Radicals
Focus
space scape: ESA, NASA, and L. Calcada (ESO for STScI); car: © Stephane Bonnel/9272855//Fotolia
Radicals not only help describe characteristics about planets, as you will see in the Module 5 Project, but they also describe a host of earthly situations, such as the skidding of a car on a wet surface when brakes are applied or the growth of a wildlife population in an animal study.
In this lesson you will refresh your knowledge and increase your ability to work with radicals. You will deal with radical expressions and equations that match situations on Earth and throughout the universe.
Outcomes
At the end of this lesson you will be able to
- express an entire radical with a numerical radicand as a mixed radical
- express a mixed radical with a numerical radicand as an entire radical
- compare and order radical expressions with numerical radicands in a given set
Lesson Questions
You will investigate the following questions:
- How do you convert between a mixed radical with a numerical radicand and an entire radical?
- Why are radicals expressed in different ways?
Assessment
Your assessment may be based on a combination of the following tasks:
- completion of the Lesson 1 Assignment (Download the Lesson 1 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
Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.
Materials and Equipment
You will need a graphing calculator.
1.1. Launch
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
- describe the terms associated with radicals
- express the square root of a perfect square as a simple number
- express a number as a square root
1.2. Are You Ready?
Module 5: Radicals
Are You Ready?
Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.
- What is the term under the radical symbol called? What is the index of a radical? Answer
- Provide the square root of the following:
- Solve the following.
- Define real number. 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
Module 5: Radicals
Refresher
To review terms associated with square roots and radicals, visit Radical. You will find a definition of radical and examples.
Khan Academy CC BY-NC-SA 3.0
To review how to express the square root of a perfect square as a simple number, watch “Understanding Square Roots.”
You can visit Square Root to find a definition of square root as well as a demonstration applet.
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
Module 5: Radicals
Discover
iStockphoto/Thinkstock
In this section you will
- investigate if or when it is possible to take the root of a negative number
- develop rules about taking the root of a negative number
Try This 1
You will use a calculator to take various roots of +2000 and −2000. 
If you prefer, you may use an applet or spreadsheet to do these calculations.
Complete Radicand Chart.
Save your work in your course folder.
Share 1
Based on your observations from Try This 1, discuss the following questions with a partner or group.
- When is it possible to take the root of a negative number? Describe the patterns you see.
- How is this related to the multiplication of negative integers?
- Summarize your discussion by creating a general rule about taking the root of a negative number.
Place a copy of this rule in your course folder.
Finding Roots of Numbers Using a Spreadsheet or Applet
If you prefer, you may use a spreadsheet to do these calculations. For directions on how to calculate a square root using a spreadsheet program, enter the keywords “taking roots spreadsheet” into an Internet search engine.
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).
So you don’t have 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.
Taking Roots with a Calculator
If your calculator has a “^” button, it means raise the preceding number to a power. Roots are rational powers. To take a cube root, raise a number to the exponent 
. For a fourth root, raise a number to the exponent 
. Enclose negative numbers and fractions in parentheses as shown.
If your calculator has an “xy” button, enter the number, press the “xy” button, and then enter the fractional exponent, in brackets, as 
. For example, if asked for the 5th root, enter the power as 
.
1.5. Explore
Module 5: Radicals
Explore
NASA, ESA, and G. Bacon (STScI)
Watch “Formalhaut Orbit Concept.” The animation shows the planet Fomalhaut b (red dot) circling its star, Fomalhaut (white dot), and the dust and debris cloud surrounding them. Notice that the planet moves faster when close to the star and slower when farther away. This motion can be predicted and described by radical expressions.
Save Module 5 Glossary Terms in your course folder now.
Here are some of the words you will want to define in Module 5 Glossary Terms in this lesson:
- radical
- radicand
- index
- mixed radical
- entire radical
1.6. Explore 2
Module 5: Radicals
Try This 2
Consider the radical 
.
- Write the radicand as a product of two factors,
, one of which is a perfect square. 
- Simplify the result by evaluating the square root of the perfect square.
- Use your calculator to verify that the result is equal to
.
- What other ways could you simplify
?
Save your responses in your course folder.
Alternatives for Simplifying Entire Radicals
When an entire radical with a numerical radicand contains one or more perfect squares, there are two methods that can be used to express the radical as a mixed radical: a product of an integer and a radical. You may have used one of these methods in Try This 2.
Method 1: Factor Out the Greatest Perfect Square
Method 2: Completely Factor the Radicand into Prime Factors
In Discover you compared taking a root of a negative and of a positive radicand. “Simplifying Radicals” reviews the reason why you can take a root of a negative number if the index number is “odd.” How does this explanation compare with the general rule you created in Share 1?
Khan Academy CC BY-NC-SA 3.0
The video also reviews how factoring the radicand helps to find the root of a large number without using a calculator. You can compare the strategy shown in the video with the one you used in Try This 2. Watch “Simplifying Radicals” now.
Self-Check 1
- Change the following entire radicals into mixed radicals using one method. Then check your answer using the other method. Which method works better for you?
- For what values of the variables in question 1 parts c. and d. do the radicals represent real numbers? Answer
Try This 3
Now try the reverse process and change a mixed radical to an entire radical. Consider the radical 
.
- Write the coefficient as a square root.
- Multiply the two radicands together.
- Use your calculator to verify that the result is equal to
.
?
is a perfect square since 
.1.7. Explore 3
Module 5: Radicals
Example: Changing a Mixed Radical to an Entire Radical
Change 
to an entire radical.
Step 1: Begin by expressing the coefficient, 6, as a square root because the radical is a square root. Square 6 and put the resulting number under a square-root sign.
Step 2: Check by computing the values for the original question and the answer using a calculator.
Note that only positive radicands are used in the examples because square roots of negative radicands would not be real numbers, as you demonstrated in the Discover section. Negative radicands, however, could be used for radicals with an odd index number, such as cube roots.
Self-Check 2
Convert the following mixed radicals into entire radicals. Verify by using a calculator to compute the numerical value of both the question and the answer. For what values of the variables in questions 2, 3, and 4 do the radicals represent real numbers?
Arranging Radicals by Size
Sometimes it will be necessary to compare and sort radicals from smallest to largest or vice versa. One method of completing this task accurately is to convert all the radicals to entire radicals. Once this is done, you only need to consider the radicands as you sort the radicals.
Read through “Example 3” on page 276 for an explanation on how this is done.
Self-Check 3
Complete “Your Turn” on page 276 of the textbook. Answer
The radical sign represents the positive root of a number. Even though (−4)2 = 16, 
. If the negative root is wanted, it must be specified as 
. If both roots are wanted, they should be specified as 
.
Skip forward to Connect if you feel you have a solid understanding of how to
- express an entire radical with a numerical radicand as a mixed radical
- express a mixed radical with a numerical radicand as an entire radical
- compare and order radical expressions in a given set
If you need a bit more practice, complete Self-Check 4.
Self-Check 4
Complete any or all of questions 3, 4, 5, 6 and 7 on pages 278 and 279 in 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.
1.8. Connect
Module 5: Radicals
In the Lesson 1 Assignment you will demonstrate your understanding of the lesson outcomes. You will apply the concepts and strategies you learned to new situations. You must show your work to support your answers.
Open your copy of Lesson 1 Assignment, which you saved in your course folder at the beginning of this lesson. Complete the assignment.
Save your work in your course folder.
Project Connection
A potentially habitable exoplanet, Gliese 581 g, has been discovered. Find out more by searching “NASA Gliese.”
You will be using radical expressions to design a habitable planet in Module 5 Project: your project. To get started, read the following sections in Module 5 Project: Plan a Planet:
- Introduction
- Assessment
Going Beyond
Find out more about Fomalhaut b and other exoplanets by searching “NASA fomalhaut.” What other telescopes are being used besides the Hubble? What other planets have recently been discovered? Look for and read related articles.
1.9. Lesson 1 Summary
Module 5: Radicals
Lesson 1 Summary
© HOWARD/3989675/Fotolia
In this lesson you investigated the following questions:
- How do you convert between a mixed radical with a numerical radicand and an entire radical?
- Why are radicals expressed in different ways?
You refreshed your knowledge of how to express an entire radical as a mixed radical. You used two different methods.
- Method 1: Factoring out the greatest perfect square
- Method 2: Completely factoring the radicand into prime factors
Both methods worked, so you can choose whichever method suits the problem and your preference.
At the beginning of this lesson you learned that real roots do not exist for negative radicands with even indexes, such as 
, 
, and 
. Real roots do exist, however, for negative radicands with odd indexes, such as 
, 
, and 
. The radical 
is only defined as a real number if x is greater than or equal to zero.
You also refreshed your knowledge about how to express a mixed radical as an entire radical. You began by expressing the coefficient as a root to the power of the index of the radical.
You checked the accuracy of your conversions by computing the mathematical values of the question and the answer with a calculator, and ensuring they were the same. You used your knowledge of converting between mixed and entire radicals to order sets of radicals from least to greatest.
In the next lesson you will enhance your ability to work with radicals, including adding, subtracting, multiplying, dividing, and simplifying radicals and their components.