NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Lesson 22: Multiplying and Dividing Expressions with Radicals Student Outcomes . Students multiply and divide expressions that contain radicals to simplify their answers. Students rationalize the denominator of a number expressed as a fraction. Lesson Notes Exercises 1–5 and the Discussion are meant to remind students of what they learned about roots in Grade 8 and Algebra I. In Grade 8, students learned the notation related to square roots and understood that the square root symbol automatically denotes the positive root (Grade 8, Module 7). In Algebra I, students used both the positive and negative roots of a number to find the location of the roots of a quadratic function. Because of the upcoming work with special triangles in this module, this lesson reviews what students learned about roots in Grade 8 Module 7 Lesson 4. For 1 √2 example, students need to understand that when they are writing the trigonometric ratios of right triangles. To √2 2 achieve this understanding, students must learn how to rationalize the denominator of numbers expressed as fractions. It is also important for students to get a sense of the value of a number. When a radical is in the denominator or is not simplified, it is more challenging to estimate its value, for example, √3750 compared to 25√6. For students who are struggling with the concepts of multiplying and dividing expressions with radicals, it may be necessary to divide the lesson so that multiplication is the focus one day and division the next. This lesson is a stepping- stone, as it moves students toward an understanding of how to rewrite expressions involving radical and rational exponents using the properties of exponents (N.RN.A.2), which are not mastered until Algebra II. The lesson focuses on simplifying expressions and solving equations that contain terms with roots. By the end of the lesson, students should understand that one reason the denominator of a number expressed as a fraction is rationalized 3 is to better estimate the value of the number. For example, students can more accurately estimate the value of √3 3√3 when written as or simply √3. Further, putting numbers in this form allows students to more easily recognize when 3 1 numbers can be combined. For example, if adding √3 and , students may not recognize that they can be combined √3 1 √3 √3 4√3 until is rewritten as . Then, the sum of √3 and is . As a teacher, it is easier to check answers when there is √3 3 3 3 an expected standard form such as a rationalized expression. Lesson 22: Multiplying and Dividing Expressions with Radicals 349 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Classwork Exercises 1–5 (8 minutes) The first three exercises review square roots that are perfect squares. The last two exercises require students to compare the value of two radical expressions and make a conjecture about their relationship. These last two exercises exemplify what is studied in this lesson. Students may need to be reminded that the square root symbol automatically denotes the positive root of the number. Scaffolding: Exercises 1–5 . Some students may need to review the perfect Simplify as much as possible. squares. A reproducible 1. √ퟏퟕퟐ = √ퟏퟕퟐ = ퟏퟕ sheet for squares of 1 30 numbers – is provided at the end of the lesson. ퟏퟎ √ퟓퟏퟎ = √ퟓퟐ × √ퟓퟐ × √ퟓퟐ × √ퟓퟐ × √ퟓퟐ 2. √ퟓ = . Consider doing a fluency = ퟓ × ퟓ × ퟓ × ퟓ × ퟓ activity that allows = ퟓퟓ students to learn their perfect squares up to 30. This may include choral 3. √ퟒ풙ퟒ = √ퟒ풙ퟒ = √ퟒ × √풙ퟐ × √풙ퟐ recitation. = ퟐ × 풙 × 풙 . English language learners = ퟐ|풙|ퟐ may benefit from choral practice with the word 4. Complete parts (a) through (c). radical. a. Compare the value of √ퟑퟔ to the value of √ퟗ × √ퟒ. The value of the two expressions is equal. The square root of ퟑퟔ is ퟔ, and the product of the square roots of ퟗ and ퟒ is also ퟔ. b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 풂 and 풃, √풂풃 = √풂 ∙ √풃. Explain. Answers will vary. Students should say that the statement √풂풃 = √풂 ∙ √풃 is valid because of the problem that was just completed: √ퟑퟔ = √ퟗ × √ퟒ = ퟔ. c. Does your conjecture hold true for 풂 = −ퟒ and 풃 = −ퟗ? No. The conjecture is not true when the numbers are negative because we cannot take the square root of a negative number. √(−ퟒ)(−ퟗ) = √ퟑퟔ = ퟔ, but we cannot calculate √−ퟒ × √−ퟗ in order to compare. 5. Complete parts (a) through (c). ퟏퟎퟎ √ퟏퟎퟎ a. Compare the value of √ to the value of . ퟐퟓ √ퟐퟓ ퟏퟎퟎ The value of the two expressions is equal. The fraction simplifies to ퟒ, and the square root of ퟒ is ퟐ. The ퟐퟓ ퟏퟎ square root of ퟏퟎퟎ divided by the square root of ퟐퟓ is equal to , which is equal to ퟐ. ퟓ Lesson 22: Multiplying and Dividing Expressions with Radicals 350 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 풂 and 풃, 풂 √풂 when 풃 ≠ ퟎ, √ = . Explain. 풃 √풃 풂 √풂 Answers will vary. Students should say that the statement √ = is valid because of the problem that was 풃 √풃 ퟏퟎퟎ √ퟏퟎퟎ just completed: √ = = ퟐ. ퟐퟓ √ퟐퟓ c. Does your conjecture hold true for 풂 = −ퟏퟎퟎ and 풃 = −ퟐퟓ? No. The conjecture is not true when the numbers are negative because we cannot take the square root of a −ퟏퟎퟎ √−ퟏퟎퟎ negative number. √ = √ퟒ = ퟐ, but we cannot calculate in order to compare. −ퟐퟓ √−ퟐퟓ Discussion (8 minutes) Debrief Exercises 1–5 by reminding students of the definition for square root, the facts given by the definition, and the rules associated with square roots for positive radicands. Whenever possible, elicit the facts and definitions from students based on their work in Exercises 1–5. Within this Discussion is the important distinction between a square root of a number and the square root of a number. A square root of a number may be negative; however, the square root of a number always refers to the principle square root or the positive root of the number. Definition of the square root: If 푥 ≥ 0, then √푥 is the nonnegative number 푝 so that 푝2 = 푥. This definition gives us four facts. The definition should not be confused with finding a square root of a number. For example, −2 is a square root of 4, but the square root of 4, that is, √4, is 2. Consider asking students to give an example of each fact using concrete numbers. Sample responses are included below each fact. Fact 1: √푎2 = 푎 if 푎 ≥ 0 √122 = 12, for any positive squared number in the radicand Fact 2: √푎2 = −푎 if 푎 < 0 This may require additional explanation because students see the answer as “negative 푎,” as opposed to the opposite of 푎. For this fact, it is assumed that 푎 is a negative number; therefore, −푎 is a positive number. It is similar to how students think about the absolute value of a number 푎: |푎| = 푎 if 푎 > 0, but |푎| = −푎 if 푎 < 0. Simply put, the minus sign is just telling students they need to take the opposite of the negative number 푎 to produce the desired result, that is, −(−푎) = 푎. √(−5)2 = 5, for any negative squared number in the radicand Fact 3: √푎2 = |푎| for all real numbers 푎 √132 = |13|, and √(−13)2 = | − 13| Fact 4: √푎2푛 = √(푎푛)2 = 푎푛 when 푎푛 is nonnegative 8 √716 = √(78)2 = 7 Lesson 22: Multiplying and Dividing Expressions with Radicals 351 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Consider asking students which of the first five exercises used Rule 1. When 푎 ≥ 0, 푏 ≥ 0, and 푐 ≥ 0, then the following rules can be applied to square roots: Rule 1: √푎푏 = √푎 ∙ √푏 A consequence of Rule 1 and the associative property gives us the following: √푎푏푐 = √푎(푏푐) = √푎 ⋅ √푏푐 = √푎 ∙ √푏 ∙ √푐 푎 √푎 Rule 2: √ = when 푏 ≠ 0 푏 √푏 We want to show that √푎푏 = √푎 ∙ √푏 for 푎 ≥ 0 and 푏 ≥ 0. To do so, we use the definition of square root. Consider allowing time for students to discuss with partners how they can prove Rule 1. Shown below are three proofs of Rule 1. Share one or all with the class. The proof of Rule 1: Let 푝 be the nonnegative number so that 푝2 = 푎, and let 푞 be the nonnegative number so that 푞2 = 푏. Then, √푎푏 = 푝푞 because 푝푞 is nonnegative (it is the product of two nonnegative numbers), and (푝푞)2 = 푝푞푝푞 = 푝2푞2 = 푎푏.