Lesson 4: Mixed and Entire Radicals

Module 2: Roots and Powers

Lesson 4: Mixed and Entire Radicals

Focus

In this lesson you will learn how to simplify radicals. You use radicals when you apply the Pythagorean theorem. In Module 1 you solved surface area and volume problems that involved the use of radicals. Solving other equations in mathematics may also involve radicals.

Like the irrational numbers you learned about in Lesson 3, radicals can be used in the creation of secret codes. In fact, a cipher has been developed to improve the security of e-mails. A computer scientist named Michael Rabin developed a system that uses the squaring of numbers to convert plaintext to ciphertext to keep the contents secret. For decryption, he uses radicals!

Outcomes

At the end of this lesson, you will be able to demonstrate an understanding of radicals by

• expressing a radical as a mixed radical in simplest form
• expressing a mixed radical as an entire radical
• explaining, using examples, the meaning of the index of a radical
• ordering a set of radicals on a number line

Lesson Questions

• How are mathematical processes reversed?
• How can the properties of radicals be used to simplify and order irrational numbers?

Lesson Completion and Assessment

As you work through each lesson, complete all the questions and learning activities in your binder using paper and pencil, clearly labeling your work (they refer to this as your course folder). These include the Are you Ready, Try This, Share and Self Check questions. Check your work if answers are provided. Remember that these questions provide you with the practice and feedback that you need to successfully complete this course.
Once you have completed all of the learning activities, take the Lesson Quiz. This is the assessment for each lesson and is located under the Assess tab or by using the Quizzes link under the Activities block.

** Note – Share questions may have to be done on your own depending on your learning situation**

Module 2: Roots and Powers

Launch

This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully.

Are You Ready?

Complete these questions in your course folder (binder). If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

  1.   List all the prime numbers between 1 and 20.

  2.   What is the prime factorization of 180?

  3.   Use prime factorization to determine the square root of 324.

  4.   Use prime factorization to determine the cube root of 1000.

  5.   Complete the table by listing the squares of each natural number from 1 to 15. The first three perfect squares have been done as an example.

Once you have completed these exercises to the best of your ability, use the provided answer link to check your work.

Answers

If you feel comfortable with the concepts covered in the questions, move forward to Discover. If you experienced difficulties or want more practice, use the resources in Refresher to review these important concepts before continuing through the lesson or contact your teacher.

Refresher

You may find the following useful for reviewing these skills.

In Lesson 1 you created a chart of all the prime numbers between 1 and 100. Retrieve this chart and review the prime numbers in the first two rows.

The “Prime Factorization” applet demonstrates how you can use a factor tree to express a number as a product of primes. Use the applet to get the prime factorization of numbers such as 324 and 1000.

Read

Go to the textbook and work through the following examples to see how prime factorization is used to identify numbers as perfect squares or perfect cubes. This method also helps you determine the square roots and cube roots of those numbers. As you read, focus on how the prime factors are regrouped into identical groups to find the square and cube roots. 

Go to “Exploring Square Roots” to explore perfect squares and non-perfect squares. If you are not sure what the difference is, then you should really check this out. You can get the most out of this applet by answering the questions that are posed. The feedback will help you gain a better understanding of the concepts presented.

Module 2: Roots and Powers

Discover

In this lesson you will learn how to convert radicals from one form to the other. In order to do so, you need to know how to manipulate radicals. This activity will help you to become more comfortable working with radicals.

Try This 1 - 2

Print out (or copy by hand) and complete the following Try This 1 and 2 activities in your course folder ( binder).
TT1 and 2 Activities
Once you complete these activities, check your work using the link below.

TT1_2 (This 1 and 2) Solutions

Module 2: Roots and Powers

Explore

The word radical has both Latin and Greek origins. From Latin, it means “root” and in Greek it means “branch.” In Lesson 2 you discovered ways to calculate the square root of a perfect square and the cube root of a perfect cube. In this lesson you will learn how to work with the roots of numbers that are not perfect squares or perfect cubes.

Glossary Terms

Add the following suggested glossary terms:

• entire radical
• index
• mixed radical
• radical
• radicand
• root

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Recall your work with the volumes of cubes. If you know the volume of a cube is 27 cm3, then you can determine the length of one side by using the formula V = s3. Here, V is the volume of the cube and s is its side length.

V = s3

    27 = s3

To obtain the answer, you would cube root both sides of the equation. (Again, you may want to use your Perfect Squares and Perfect Cubes Data Sheet you created in Lesson 2 and saved in your course folder.)

In this example you determined the cube root of 27 to find the side length of the cube. The expression is called a radical.

Study the diagram to understand how to name the parts of a radical.

The radicand is the number or expression that appears underneath the radical symbol.

The index identifies the type of root. For example, a square root has an index of 2. A cube root has an index of 3. A fourth root has an index of 4, and so on.

Sometimes a radical may have a coefficient, or a number in front of the radical symbol, separated from the radicand. A radical whose coefficient is a number other than 1 is called a mixed radical. On the other hand, a radical with a coefficient equal to 1 is called an entire radical. The following chart has examples of mixed radicals and entire radicals.

In Lesson 2 you learned how to find the square and cube roots of perfect squares and perfect cubes.

What if you are asked to find the square root of a non-perfect square such as 12? Well, you might say that is an irrational number. You could find a decimal approximation, but the only way to get an exact value is by simplifying the radical.

In the Discover section you found out the square root of the product of two numbers is equal to the product of the square roots of those numbers. This property can be summarized with the following expression:

where a and b are positive numbers

For instance, the expression can be evaluated as.

The following example uses this concept to simplify.

Example 1

Simplify .

Solution

The radical could be expressed as the product of square roots in several ways:

Sometimes, a radical is not a perfect square, but it may have a perfect square amongst its factors. To simplify the radical, you need to rewrite the radicand as a product of two factors, one of which is a perfect square. You can then simplify the square root of the perfect square and reduce the radicand. This process may need to be repeated until the radicand can no longer be reduced.

Example 2

Simplify the radical

Solution

Method 1: Perfect Square Factors

or look for the largest perfect square and get

Method 2: Prime Factorization

Watch and Listen

Watch “Simplifying Radicals” that demonstrates the method by prime factorization.

Did you notice the result of simplifying an entire radical, is a mixed radical,

Share

Compare your answers to the following questions with another classmate or appropriate partner ( if possible).
1. In Method 1, how can you avoid repeated factoring of the radicand?
2. In Method 2, how do pairs of identical factors become simplified? Can you show the math that makes this possible?

Record any similarities and differences you discussed in your course folder (binder).

Possible Answers

Example 3

You can apply the same techniques to simplify radicals of different indices (plural form of index).

Solution

Method 1: Perfect Cube Factors

Method 2: Prime Factorization

Module 2: Roots and Powers

Read

Go to the textbook and work through the examples to further see how this is done. Pay attention to how the method is different from simplifying square roots.

You have learned to simplify entire radicals by converting them into reduced mixed radicals. There will be some cases, however, where it is helpful to do the opposite. Read the next example to see how you can convert mixed radicals into entire radicals.

Example 4

Write as an entire radical.

Solution

What distinguishes an entire radical from a mixed radical is that the entire radical does not have a coefficient. So, the objective is to determine the value of the radicand.

Method 1: Perfect Square Factors

The number 2 can be written as

Method 2: Prime Factorization

Remember that the coefficient of 2 came from a pair of 2s under the radical symbol.

Write 2 as

Example 5

Write as an entire radical.

Solution

Using the method of prime factorization, write 4 as

Read

At this point, if you think you understand how to convert from entire radicals to mixed radicals and vice versa, proceed to the Self-Check. Otherwise go to the textbook and work through the next example for more help with this concept.

In the previous lesson you learned how to locate irrational numbers on a number line. In this lesson you have learned to convert mixed radicals to entire radicals. Both these concepts can be applied to the ordering of irrational numbers on a number line. Study the following example to see how each method is applied.

Example 6

Order these irrational numbers from least to greatest:

Solution

Method 1: Number Line

Enter the radicals into your calculator to obtain a decimal approximation of each one.

Putting these on a number line will give you the correct order:

Method 2: Conversion to Entire Radicals

Convert each of the mixed radicals into entire radicals:

Since each of the radicals has the same index, you only need to compare the radicands.

The correct order from least to greatest is

Self-Check

SC 1. Simplify

SC 2. Rewrite as a mixed radical.

SC 3. Express as an entire radical.

SC 4. Convert to an entire radical.

Compare your answers.

Try This 3

Put your knowledge to the test by working through some of the questions from your textbook. When you are converting a mixed radical to an entire radical, pay attention to the index so you know how many factors of the coefficient you need to multiply with the radicand. When you are converting an entire radical, take an extra moment to make sure you have expressed the radicand as a product using the largest possible perfect square as a factor. This will help you avoid repeated multiplication.

Complete the following in your course folder ( binder).

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 3. Complete “Exercises” questions 4, 5, 11.a), 11.b), 11.c), 11.d), 12.a), 12.b), 12.c), and 12.d) on page 218.

Use the link below to check your answers to Try This 3.

Possible TT3 (Try This 3) Solutions

Module 2: Roots and Powers

Complete the lesson quiz posted under the Assess tab or by using the Quizzes link under the Activities block. Also,  ensure your work in your binder (course folder) is complete.

Project Connection ** NOT ASSIGNED**

There is no project component in this lesson. You will continue working on your project in the final two lessons of this module.

Going Beyond

Before the advent of calculators, the evaluation of square roots was performed using paper-and-pencil methods.

Step 1: Start an Internet search and use the keywords “Newton’s method square root” or “Babylonian method” to discover the steps for finding square roots without a calculator.

Step 2: Choose one of these methods and write down the steps.

Step 3: Apply the method to the following questions:

1. Evaluate
2. Evaluate

Module 2: Roots and Powers

 Lesson 4 Summary

No one ever wants to be involved in a collision! Suppose you know the distance you need to stop your car, but can you stop it at the speed you are travelling? A very important application of radicals helps you find the maximum speed of a car to stop in a certain distance. The experimentation with objects in motion has shown that a relationship exists among the variables. For example, when you double your speed, your stopping distance quadruples.

Physicists have worked to create a formula that shows speed and stopping distance

In this radical expression, clear evidence is shown for why drivers should slow down in urban areas; a small reduction in speed leads to a much larger reduction in stopping distance. Solving the radical equation correctly here could quite literally save your, or someone else’s, life!

In Lesson 4 you investigated the following questions:

• How are mathematical processes reversed?
• How can the properties of radicals be used to simplify and order irrational numbers?

In this lesson you learned to identify the parts of a radical. You learned that an index of 2 means you can take one of two equal factors out from under the radical symbol. In a similar way, an index of 3 means you can take one of three equal factors out from under the radical symbol. By converting mixed radicals of the same index into entire radicals, you learned that you can compare the radicands to determine the correct sequence when ordering irrational numbers.

When you are encrypting a message, it is critical for you to be able to decrypt the code at a later time. Decryption is the reverse process of encryption. In a previous lesson you learned that you can reverse the single-step mathematical process of squaring by square rooting. Similarly, you can reverse the cubing of a number by evaluating the result’s cube root. In this lesson you examined another kind of mathematical process and its reverse process—converting between mixed radicals and entire radicals. Converting from an entire radical to a mixed radical is a two-step process. In the case of radicals with an index of 2, you first have to express the radicand as a product of two factors—a perfect square and a non-perfect square—and then you square root each factor. To reverse this you have to not only reverse each step but also in the reverse order.

Compare the following:

Notice that to convert to a mixed radical, you need to divide and then square root. On the other hand, to convert back to an entire radical, you need to square and then multiply. Therefore, each step is reversed, and the order of these steps is also reversed. The reversibility of processes is a recurring concept in mathematics. You will see this again in future courses when you examine inverse relations.

In the next lesson you will review the exponent laws. You will see by the last lesson of this module how radicals and exponents are related to each other. You will also learn how to convert between radicals and powers with rational exponents.