Algebra Simplifying Radicals StAIR Project Lori Ferrington

Objectives and Standards Simplifying Radicals Michigan Department of Education - High School Content Expectations Students will know the properties of positive and negative roots. (A1.1.2)  Students will know how to simplify positive and negative radicals for later use in algebraic equations. (A1.1.2) Students will know how to multiply radical expressions for later use in algebraic equations. (A1.1.2)

Introduction Simplifying Radicals By clicking on this icon, you can access the definitions of key terms at any time during this lesson. This lesson is designed for Mrs. Ferrington’s Algebra class. You are to navigate your way through this lesson individually. The home page will allow you to navigate through different parts of this lesson If you link out to a webpage, close the window when you are done viewing the content on the webpage to return to this lesson. Key terms in this lesson will be shown in red. After this lesson, you will be given an assessment to measure your understanding. Have Fun! :) Click to return to the previous page Click to return to the home page Click to go on to the next page

Let’s review factor trees and prime factorization. Warm up Let’s review factor trees and prime factorization. A prime number is only divisible by 1 and itself. A factor is a number that can divide another number without a remainder. 2 and 3 are prime factors of 6 because 23=6 and both 2 and 3 are prime. 180 10 18 5 2 3 6 3 2 So, we can say that the prime factorization of 180 is 22335.

Need extra practice with factor trees? Warm Up Need extra practice with factor trees? Video explanation here Extra practice here You think you’re ready…click the next button to continue.

Complete the factor tree below. Warm UP Question #1 Complete the factor tree below. Which of the following are factors of 252? A.) 2126 252 B.) 2118 C.) 386 D.) 551

Complete the factor tree below. WARM up Question #1 Complete the factor tree below. Great job! 2 and 126 are factors of 252. Which of the following are factors of 126? 252 A.) 264 2 126 B.) 267 C.) 914 D.) 719

Complete the factor tree below. Warm Up Question #1 Complete the factor tree below. 2 and 126 are factors of 252. Fabulous! 9 and 14 are factors of 126. Which of the following are factors of 9? 252 2 126 A.) 23 9 14 B.) 33 C.) 22 D.) 34

Complete the factor tree below. Warm Up Question #1 Complete the factor tree below. 2 and 126 are factors of 252. 9 and 14 are factors of 126. Excellent! 3 and 3 are factors of 9. Which of the following are factors of 14? 252 2 126 9 14 A.) 23 3 B.) 25 C.) 37 D.) 27

Complete the factor tree below. Warm Up Question #1 Complete the factor tree below. 2 and 126 are factors of 252. 9 and 14 are factors of 126. 3 and 3 are factors of 9. Awesome! 2 and 7 are factors of 14. 252 2 126 9 14 3 7 2 The prime factorization of 252 is 22337. Great job! Click the next arrow to try another.

Find the prime factorization of 112. Warm up question #2 Find the prime factorization of 112. Some common prime numbers are 2, 3, 5, 7, 11, and 13. A.) 2357 112 B.) 22227 C.) 2247 D.) 23335

Find the prime factorization of 112. Great Job! Warm up question #2 Find the prime factorization of 112. You’re right!! 112 22227 = 112 2 56 8 7 4 2 2 Now let’s move on to some new stuff.

III. Product of Two Radicals Home Simplifying Radicals Home A radical is any expression that contains a square root, cube root, etc. The symbol representation of a radical is √ . The term radical comes from the late Latin radicallis meaning “of roots” and from Latin radix meaning “root.” We say that a radical expression is simplified, or in its simplest form, when the radicand has no square factors. There are three parts to the following lesson that will teach you about simplifying radicals. Please follow them in order. You will be able to navigate back to review any lessons again. I. Positive Radicals II. Negative Radicals III. Product of Two Radicals Quiz

i. Positive radicals Example #1: Simplify √12 Check for a perfect square Find the prime factorization of the radicand Identify pairs of primes in radicand Simplify perfect squares √12 = 3.4641; this is not a perfect square. √12 = √223 = √22  √3 = √4  √3 = 2√3 12 2 6 3 So √12 simplifies to 2√3.

I. positive radicals Example #2: Simplify √162 √162 = 12.7279; this is not a perfect square. 162 2 81 9 3 √162 = √23333 = √2  √33  √33 = √2  √9  √9 = 33 √2 = 9√2 Be sure that any perfect squares come out in front (to the left) of the radical symbol. This prevents confusing it with numbers in the radicand.

Check for Understanding… i. Positive radicals Check for Understanding… Simplify 2√605 Hint: If an integer is in front of the radical, do not move it. A.) 11√5 B.) 11√2 C.) 10√11 D.) 22√5 For extra examples go here

Check for Understanding… Try Again! i. Positive radicals Check for Understanding… Oops, 2√605 ≠ 11√5 Don’t forget the integer in front of the radical symbol

Check for Understanding… Try Again! i. Positive radicals Check for Understanding… Sorry, 2√605 ≠ 11√2 You can only square root perfect squares. Look for pairs of numbers in the radicand.

Check for Understanding… Try Again! i. Positive radicals Check for Understanding… Bummer, 2√605 ≠ 10√11 You can only square root perfect squares. Look for pairs of numbers in the radicand.

Check for Understanding… Great Job! i. Positive radicals Check for Understanding… Yes!! 2√605 = 2√51111 = 2 √5  √1111 = 2 √5  √121 = 211 √5 = 22√5 Now on to negative radicals in section II …

What do you think happens if a negative is outside of the radical? ii. negative radicals What do you think happens if a negative is outside of the radical? Let’s look at example #1: Simplify -2√72 Which do you think is the correct answer? Hint: many of the same properties of simplifying positive radicals apply in this situation. A.) 4√6 B.) -12√2 C.) 12√3 D.) -8√3

What do you think happens if a negative is outside of the radical? II. negative radicals What do you think happens if a negative is outside of the radical? Example #1: Simplify -2√72 You’re correct! The answer is B.) -12√2, but why? What rule best fits when simplifying radicals with a negative in front of them? A.) A negative in front of the radical goes in the radicand. B.) A negative in front of the radical stays in front. C.) A negative in front of the radical cannot be simplified.

II. Negative radicals Example #2: Simplify -5√50 √50 = 7.017; this is not a perfect square. Don’t forget the rule you determined in the previous slide: A negative in front of the radical stays in front. 50 2 25 5 -5√50 = -5√255 = -5  √2  √55 = -5  √2  √25 = -55 √2 = -25√2

Check for understanding… II. Negative Radicals Check for understanding… Simplify -2√98 A.) -14√2 B.) -7√2 C.) -2√7 D.) -4√7 For extra examples go here

Check for Understanding… Try Again! iI. Negative radicals Check for Understanding… Oh no, -2√98 ≠ -7√2 Don’t forget about the integer in front of the radical

Check for Understanding… Try Again! iI. Negative radicals Check for Understanding… Sorry, -2√98 ≠ -2√7 Look for pairs of numbers in the radicand.

Check for Understanding… Try Again! iI. Negative radicals Check for Understanding… Nice try, -2√98 ≠ -4√7 Look for perfect squares in the radicand.

Check for Understanding… Fantastic! iI. Negative radicals Check for Understanding… You got it!! -2√98 = -2  √277 = -2 √2  √77 = -2 √2  √49 = -27 √2 = -14√2 Let’s look at the multiplying two radicals in section III …

III. Product of 2 radicals Example #1: Simplify 2√3  √12 Check for perfect squares. Multiply the numbers in front of the radical and multiply the radicands. Simplify the radicand by finding the factorization or by identifying perfect squares Neither √3 nor √12 are perfect squares. 2√3  √12 = 2 √312 = 2 √36 = 26 = 12 So 2√3  √12 simplifies to 12.

III. Product of 2 radicals Example #2: Simplify 3√3  2√8 Neither √3 nor √8 are perfect squares. 3√3  2√8 = 32 √38 = 6 √24 = 6 √2223 = 6 √22  √23 = 6  √4  √6 = 6  2 √6 = 12√6 Remember to use factor trees to help you find the prime factorization of the radicand.

III. Product of 2 radicals Check for understanding… Simplify √10  √5 A.) 5√5 B.) 2√5 C.) -2√5 D.) 5√2 For extra examples go here

III. Product of 2 radicals Try Again! III. Product of 2 radicals Check for Understanding… Oh no, √10  √5 ≠ 5√5 Look for perfect squares in the radicand.

III. Product of 2 radicals Try Again! III. Product of 2 radicals Check for Understanding… Not that one, √10  √5 ≠ 2√5 Look for perfect squares or pairs of integers in the radicand.

III. Product of 2 radicals Try Again! III. Product of 2 radicals Check for Understanding… Whoops, √10  √5 ≠ -2√5 Be careful of your use of negatives.

III. Product of 2 radicals Awesome! III. Product of 2 radicals Check for Understanding… That’s correct! √10  √5 = √105 = √50 = √255 = √2  √55 = √2  √25 = 5√2 Click next to take the Simplifying Radicals Quiz …

Click here to begin the quiz Instructions Move through this quiz by selecting the correct simplified form of the radical expression given. You must get each problem correct before proceeding to the next question. Good luck! Click here to begin the quiz Don’t forget your factor trees! 

1 QUIZ 1. Simplify √200 A.) 20√2 B.) 10√5 C.) 10√2 D.) 2√5

Oops, go back and try again! QUIZ 1. Simplify √200 Hint: Find your prime factorization of 200, and then look for perfect squares. Oops, go back and try again!

2 Quiz 2. Simplify 10√1000 A.) 10√10 B.) 100√10 C.) 10√100 D.) 20√10

Oops, go back and try again! Quiz 2. Simplify 10√1000 Hint: Don’t forget the 10 in front of the radical. Oops, go back and try again!

3 Quiz 3. Simplify -√32 A.) -4√8 B.) 4√4 C.) -8√2 D.) -4√2

Oops, go back and try again! Quiz 3. Simplify -√32 Hint: Remember the rule about negatives in front of the radical symbol. Oops, go back and try again!

4 Quiz 4. Simplify -3√27  4√3 A.) -108 B.) -36√3 C.) -81 D.) -√93

Oops, go back and try again! Quiz 4. Simplify -3√27  4√3 Hint: Multiply your integers in front of the radical and the multiply the radicands before you simplify. Oops, go back and try again!

Great! You did it! Fabulous! Superb Excellent

Terms and definitions (in order of appearance) Vocabulary Terms and definitions (in order of appearance) Prime number Factor Prime Factorization Radical Radicand Perfect Square A number only divisible by only 1 and itself A number that can divide another number without a remainder. The list of all of the prime numbers whose product makes up a number. Any expression that contains a root (e.g. square root). The symbol is √ The number underneath the root or radical symbol. It is the product of some integer with itself (e.g. √9 = 3).