chap11 bk

Full text

(1)Notes Introduction In this unit, students will be introduced to additional nonlinear functions such as radical and rational equations. Students learn how to simplify radical and rational expressions, and how to solve equations involving these expressions. Students also explore triangles through the Pythagorean Theorem and trigonometric ratios.. Nonlinear functions such as radical and rational functions can be used to model real-world situations such as the speed of a roller coaster. In this unit, you will learn about radical and rational functions.. Assessment Options Unit 4 Test Pages 779–780 of the Chapter 12 Resource Masters may be used as a test or review for Unit 4. This assessment contains both multiple-choice and short answer items.. TestCheck and Worksheet Builder This CD-ROM can be used to create additional unit tests and review worksheets.. 582 Unit 4 Radical and Rational Functions. 582. Unit 4 Radical and Rational Functions. Radical and Rational Functions Chapter 11 Radical Expressions and Triangles. Chapter 12 Rational Expressions and Equations.

(2) Teaching Suggestions Have students study the USA TODAY Snapshot. • Ask students how the data in the graph differs from that of a linear function. The data in the graph does not follow a straight line, therefore it is not linear. • Why might math be important to a roller coaster designer? Sample answer: A roller coaster designer might use math to calculate the speed of a roller coaster before it is built to make sure it is safe.. Building the Best Roller Coaster. USA TODAY Snapshots®. Each year, amusement park owners compete to earn part of the billions of dollars Americans spend at amusement parks. Often the parks draw customers with new taller and faster roller coasters. In this project, you will explore how radical and rational functions are related to buying and building a new roller coaster. Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 4.. A day in the park. What the typical family of four pays to visit a park1:. $163 $180. 11-1 590. 12-2 652. $116. $120 $60 0. Lesson Page. Additional USA TODAY Snapshots appearing in Unit 4: Chapter 11 Global spending on construction (p. 615) Chapter 12 Most Americans have one or two credit cards (p. 672) Cost of parenthood rising (p. 689). ’93. ’95. ’97. ’99. ’01. 1 — Admission for two adults and two children, parking for one car and purchase of two hot dogs, two hamburgers, four orders of fries, four small soft drinks and two children’s T-shirts. Source: Amusement Business. Unit 4. By Marcy E. Mullins, USA TODAY. Radical and Rational Functions 583. Internet Project A WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the next step in their WebQuest. At the end of Chapter 12, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 4 Radical and Rational Functions 583.

(3) Radical Expressions and Triangles Chapter Overview and Pacing PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. Simplifying Radical Expressions (pp. 586–592) • Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots.. 2. 2. 1.5. 1.5. Operations with Radical Expressions (pp. 593–597) • Add and subtract radical expressions. • Multiply radical expressions.. 2. 2. 1. 1. Radical Equations (pp. 598–604) • Solve radical equations. • Solve radical equations with extraneous solutions. Follow-Up: Use a graphing calculator to graph radical equations.. 2. 3 (with 11-3 Follow-Up). 1. 2. The Pythagorean Theorem (pp. 605–610) • Solve problems by using the Pythagorean Theorem. • Determine whether a triangle is a right triangle.. 1. 1. 0.5. 0.5. The Distance Formula (pp. 611–615) • Find the distance between two points on the coordinate plane. • Find a point that is a given distance from a second point in a plane.. 1. 1. 0.5. 0.5. Similar Triangles (pp. 616–621) • Determine whether two triangles are similar. • Find the unknown measures of sides of two similar triangles.. 1. 1. 0.5. 0.5. Trigonometric Ratios (pp. 622–630) Preview: Use paper triangles to investigate trigonometric ratios. • Define the sine, cosine, and tangent ratios. • Use trigonometric ratios to solve right triangles.. 2. 2. 1. 1. Study Guide and Practice Test (pp. 632–637) Standardized Test Practice (pp. 638–639). 1. 1. 0.5. 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 13. 14. 7. 8. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 584A Chapter 11 Radical Expressions and Triangles.

(4) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 643–644. 645–646. 647. 648. 649–650. 651–652. 653. 654. 655–656. 657–658. 659. 660. 661–662. 663–664. 665. 666. 667–668. 669–670. 671. 672. 673–674. 675–676. 677. 678. 700. 679–680. 681–682. 683. 684. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 11 RESOURCE MASTERS. See pages T12–T13.. SC 21. 83. 11-1. 11-1. 84. 11-2. 11-2. GCS 43. 85. 11-3. 11-3. GCS 44, SC 22. 86. 11-4. 11-4. SM 85–90. 87. 11-5. 11-5. 88. 11-6. 11-6. 700. 89. 11-7. 11-7. 685–698, 702–704. 90. 699. 37–38. 699, 701. 61–62. Materials. graphing calculator. graphing calculator (Follow-Up: graphing calculator) 32. (Follow-Up: ruler, grid paper, protractor), string, drinking straw, paper clip, protractor, tape, meter sticks or tape measure. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 11 Radical Expressions and Triangles 584B.

(5) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 2, students learned to find principal square roots. They solved proportions in Chapter 3. Students found the factors of the difference of squares in Chapter 8.. This Chapter In this chapter, students simplify and perform operations with radical expressions. They apply these skills to solving radical equations. The Pythagorean Theorem is used to solve problems involving right triangles. The Distance Formula is introduced as an application of the Pythagorean Theorem. Similar triangles and trigonometric ratios are used to find missing measures in triangles.. Future Connections The basics of trigonometry and the study of triangles are explored in this chapter. These basics will be applied to deeper studies of triangles in later mathematics courses. Many of these concepts are used in construction and engineering.. 584C. Chapter 11 Radical Expressions and Triangles. Simplifying Radical Expressions A radical expression contains a square root. The expression is in simplest form if the expression inside the radical sign, or radicand, has only 1 as a perfect square factor. The Product Property of Square Roots states that the square root of a product equals the product of each square root. Prime factorizations combined with the Product Property of Square Roots can be used to simplify radical expressions. Principal square roots are never negative, so absolute value symbols must be used to signify that some results are not negative. The Quotient Property of Square Roots states that the square root of a quotient equals the quotient of each square root. The Quotient Property of Square Roots can be used to derive the Quadratic Formula. A fraction does not have a radical in its denominator if it is in simplest form. Since squaring and taking a square root are inverse functions, you multiply the numerator and the denominator by the same number so that the denominator contains a perfect square. Remember that the numerator must be multiplied by the same amount so that the whole fraction is being multiplied by a value of 1. This process is called rationalizing the denominator. After rationalizing the denominator, check for coefficients of the radical in the numerator that will simplify with the denominator. If the denominator is an expression containing a radical, multiply by the conjugate. For example, if the denominator is a b, multiply by a b.. Operations with Radical Expressions Use the process of combining like terms to simplify expressions in which radicals are added or subtracted. For terms to be combined, their radicands must be the same. As in combining monomials with variables, only the coefficients of the radicals are combined. Be sure to simplify all radicals first. When multiplying radical expressions, first multiply the coefficients, and then use the Product Property of Square Roots to multiply the radicals. Simplify each term and then combine like terms as necessary.. Radical Equations Equations that contain variables in the radicand are called radical equations. To solve radical equations, the radical must first be isolated on one side. Then square each side. This will eliminate the radical. This process sometimes produces results that are not solutions of the original equation. These are called extraneous solutions. All solutions must be substituted back into the original equation to check their validity..

(6) The Pythagorean Theorem The two sides of a right triangle that form the right angle are the legs of the triangle. The third, and longest, side is the hypotenuse. The hypotenuse is the longest side because it is always across from the angle with the greatest measure. The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse. The formula is c2 a2 b2, where a and b are the measures of the legs and c is the measure of the hypotenuse. This formula can be used to find the length of any missing side of a right triangle if the lengths of the other two sides are known. Any three whole numbers that satisfy this equation are known as a Pythagorean Triple. These triples represent side lengths that always form right triangles. It follows that if three numbers do not satisfy the Pythagorean Theorem, then sides of their length will not form a right triangle.. Trigonometric Ratios Trigonometry is the mathematical study of angles and triangles. Ratios comparing the measures of two sides of a right triangle are called trigonometric ratios. The three most common trigonometric ratios are sine, cosine, and tangent. These ratios can be used to find the measures of missing sides or the measures of the acute angles. The relationship of the two sides necessary for the problem to a specific acute angle determines which ratio is used. If the measures of just two sides of a triangle or the measures of one side and one acute angle are known, then the measures of all of the rest of the sides and angles can be found. This is called solving the triangle. Trigonometric ratios are used to find distances in problems involving angles of elevation and angles of depression.. The Distance Formula If the Pythagorean Theorem is solved for c, the result is the Distance Formula. The variable a is expressed as the difference of the x-coordinates of the endpoints of the hypotenuse and b is expressed as the difference of the y-coordinates. This formula is used to find the distance between any two points. You can also find one missing coordinate of an endpoint if you know the other coordinate, the coordinates of the other endpoint, and the distance between the two points.. Similar Triangles Similar triangles have the same shape, but are not necessarily the same size. All of the corresponding angles will have equal measures, and the corresponding sides will all be proportional. If the sides have a 1 to 1 ratio of proportionality, then the similar triangles are the same size. When determining whether triangles are similar, all you need to check is if the corresponding angles have the same measure. If all the angle measures cannot be determined, then check the corresponding sides to see if they are proportional. Proportions can be used to find the lengths of missing sides of similar triangles. You must know the lengths of at least one pair of corresponding sides and the length of the side that corresponds to the missing side’s length. Set up the proportion, and then cross multiply. Solve the resulting equation.. Chapter 11 Radical Expressions and Triangles 584D.

(7) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 585, 592, 597, 603, 610, 615, 621 Practice Quiz 1, p. 603 Practice Quiz 2, p. 621. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 37–38, 61–62 Quizzes, CRM pp. 699–700 Mid-Chapter Test, CRM p. 701 Study Guide and Intervention, CRM pp. 643–644, 649–650, 655–656, 661–662, 667–668, 673–674, 679–680. Mixed Review. pp. 592, 597, 603, 610, 615, 621, 630. Cumulative Review, CRM p. 702. Error Analysis. Find the Error, pp. 600, 618. Find the Error, TWE pp. 600, 618 Unlocking Misconceptions, TWE p. 612 Tips for New Teachers, TWE pp. 587, 624. Standardized Test Practice. pp. 591, 597, 602, 606, 608, 610, 615, 620, 630, 637, 638–639. TWE pp. 638–639 Standardized Test Practice, CRM pp. 703–704. Open-Ended Assessment. Writing in Math, pp. 591, 597, 602, 610, 614, 620, 630 Open Ended, pp. 589, 595, 600, 607, 612, 618, 627 Standardized Test, p. 639. Modeling: TWE pp. 603, 610, 621 Speaking: TWE pp. 597, 615 Writing: TWE pp. 592, 630 Open-Ended Assessment, CRM p. 697. Chapter Assessment. Study Guide, pp. 632–636 Practice Test, p. 637. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 685–690 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 691–696 Vocabulary Test/Review, CRM p. 698. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 584E. Chapter 11 Radical Expressions and Triangles.

(8) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson 11-4. AlgePASS Lesson 32 Solving Problems Using the Pythagorean Theorem. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 585 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 589, 595, 600, 607, 612, 618, 627) • Reading Mathematics, p. 631 • Writing in Math questions in every lesson, pp. 591, 597, 602, 610, 614, 620, 630 • Reading Study Tip, pp. 586, 611, 616, 623 • WebQuest, p. 590. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 83–90 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Teacher Wraparound Edition • Foldables Study Organizer, pp. 585, 632 • Study Notebook suggestions, pp. 589, 595, 600, 608, 613, 618, 627, 631 • Modeling activities, pp. 603, 610, 621 • Speaking activities, pp. 597, 615 • Writing activities, pp. 592, 630 • Differentiated Instruction, (Verbal/Linguistic), p. 599 • ELL Resources, pp. 584, 591, 596, 599, 602, 609, 614, 620, 629, 631, 632 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 11 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 11 Resource Masters, pp. 647, 653, 659, 665, 671, 677, 683) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 3.2, 7.9, 7.10, 8.6. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 11 Radical Expressions and Triangles 584F.

(9) Radical Expressions and Triangles. Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. • Lessons 11-1 and 11-2 Simplify and perform operations with radical expressions. • Lesson 11-3 Solve radical equations. • Lessons 11-4 and 11-5 Use the Pythagorean Theorem and Distance Formula. • Lessons 11-6 and 11-7 Use similar triangles and trigonometric ratios.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. Key Vocabulary • • • • •. radical expression (p. 586) radical equation (p. 598) Pythagorean Theorem (p. 605) Distance Formula (p. 611) trigonometric ratios (p. 623). Physics problems are among the many applications of radical equations. Formulas that contain the value for the acceleration due to gravity, such as free-fall times, escape velocities, and the speeds of roller coasters, can all be written as radical equations. You will learn how to calculate the time it takes for a skydiver to fall a given distance in Lesson 11-3.. Lesson 11-1 11-2 11-3 11-3 Follow-Up 11-4 11-5 11-6 11-7 Preview 11-7. NCTM Standards. Local Objectives. 1, 2, 6, 8, 9, 10 1, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8 1, 2, 3, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10 1, 3, 7 3, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 584. 584 Chapter 11 Radical Expressions and Triangles. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 11 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 11 test.. Chapter 11 Radical Expressions and Triangles.

(10) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 11. For Lessons 11-1 and 11-4. This section provides a review of the basic concepts needed before beginning Chapter 11. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 37–38 and 61–62.. Find Square Roots. Find each square root. If necessary, round to the nearest hundredth. (For review, see Lesson 2-7.). 1. 25 5. 2. 80 8.94. 3. 56 7.48. 4. 324 18. For Lesson 11-2. Combine Like Terms. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson.. Simplify each expression. (For review, see Lesson 1-6.) 5. 3a 7b 2a a 7b. 6. 14x 6y 2y 14x 4y. 7. (10c 5d) (6c 5d) 16c. 8. (21m 15n) (9n 4m) 25m 6n. For Lesson 11-3. Solve Quadratic Equations. Solve each equation. (For review, see Lesson 9-3.) 10. x2 10x 24 0 {6, 4} 1 12. 2x2 x 1 2 , 1 2. 9. x(x 5) 0 {0, 5}. . 11. x2 6x 27 0 {3, 9}. . For Lesson 11-6. Proportions. Use cross products to determine whether each pair of ratios forms a proportion. Write yes or no. (For review, see Lesson 3-6.) 2 8 13. , 3 12. yes. 4 16 14. , 5 25. no. 8 12 15. , 10 16. 6 3 16. , 30 15. no. For Lesson. Prerequisite Skill. 11-2 11-3. Multiplying Binomials (p. 592) Finding Special Products (p. 597) Evaluating Radical Expressions (p. 603) Simplifying Radical Expressions (p. 610) Solving Proportions (p. 615) Evaluating Expressions (p. 621). 11-4. yes. 11-5 11-6 11-7. Make this Foldable to help you organize information about radical expressions and equations. Begin with a sheet of 1 plain 8" by 11" paper. 2. Fold in Half. Fold Again Fold the top to the bottom.. Fold in half lengthwise.. Cut. Label Open. Cut along the second fold to make two tabs.. Label each tab as shown.. Radical Expression s Radical Equations. As you read and study the chapter, write notes and examples for each lesson under each tab.. Reading and Writing. Chapter 11 Radical Expressions and Triangles 585. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data: Annotating As students read and work through the chapter, have them make annotations under the tabs of their Foldable. Explain to them that annotations are usually notes taken in the margins of books we own to organize the text for review or studying. Annotations often include questions that arise as we read the chapter, reader comments and reactions, sentence length summaries, steps or data numbered by the reader, and key points highlighted or underlined. Chapter 11 Radical Expressions and Triangles 585.

(11) Simplifying Radical Expressions. Lesson Notes. 1 Focus 5-Minute Check Transparency 11-1 Use as a quiz or review of Chapter 10. Mathematical Background notes are available for this lesson on p. 584C.. • Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots.. • radical expression • radicand • rationalizing the denominator • conjugate. A spacecraft leaving Earth must have a velocity of at least 11.2 kilometers per second (25,000 miles per hour) to enter into orbit. This velocity is called the escape velocity. The escape velocity of an object is given by the radical expression 2GM , where G is the gravitational constant, R. M is the mass of the planet or star, and R is the radius of the planet or star. Once values are substituted for the variables, the formula can be simplified.. Building on Prior Knowledge Students were introduced to the Quadratic Formula in Lesson 10-4 and they learned how to use it to solve quadratic equations. In this lesson, students learn how to derive the Quadratic Formula from the standard form of a quadratic equation using the Quotient Property of Square Roots. are radical expressions used in space exploration? Ask students: • In the formula for escape velocity, what does the radical sign mean? The radical sign means that you must find the square root of the value under the radical sign. • Based on what you know about the order of operations, how do you think you should simplify the radical expression in the escape velocity formula? You should simplify the expression under the radical sign before finding the square root.. are radical expressions used in space exploration?. Vocabulary. PRODUCT PROPERTY OF SQUARE ROOTS A radical expression is an expression that contains a square root. A radicand, the expression under the radical sign, is in simplest form if it contains no perfect square factors other than 1. The following property can be used to simplify square roots.. Product Property of Square Roots • Words. For any numbers a and b, where a 0 and b 0, the square root of a product is equal to the product of each square root.. • Symbols ab a b . • Example 4 25 4 25 . The Product Property of Square Roots and prime factorization can be used to simplify radical expressions in which the radicand is not a perfect square.. Example 1 Simplify Square Roots Simplify. a. 12 . Study Tip Reading Math. 23 is read two times the square root of 3 or two radical three.. 12 2 2 3 22 3 23. Prime factorization of 12 Product Property of Square Roots Simplify.. b. 90 90 23 35 32 25 310 . Prime factorization of 90 Product Property of Square Roots Simplify.. 586 Chapter 11 Radical Expressions and Triangles. Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 643–644 • Skills Practice, p. 645 • Practice, p. 646 • Reading to Learn Mathematics, p. 647 • Enrichment, p. 648. Parent and Student Study Guide Workbook, p. 83 School-to-Career Masters, p. 21. Transparencies 5-Minute Check Transparency 11-1 Answer Key Transparencies. Technology Interactive Chalkboard.

(12) The Product Property can also be used to multiply square roots.. Study Tip Alternative Method. To find 3 15 , you could multiply first and then use the prime factorization.. 45 3 15 32 5 35. 2 Teach. Example 2 Multiply Square Roots Find 3 15 .. 3 3 5 3 15 32 5 35. PRODUCT PROPERTY OF SQUARE ROOTS. Product Property of Square Roots Product Property Simplify.. Intervention In order to simplify square roots with the Product Property of Square Roots, students need to be able to find the prime factorization of the radicand. Take a few minutes to review finding prime factorizations. After a quick refresher, students can focus on learning the new concept, rather than trying to recall earlier material.. New. When finding the principal square root of an expression containing variables, be x2. It may seem that sure that the result is not negative. Consider the expression x2 x. Let’s look at x 2. x2 x 2 2 (2) 4 2 2 2. Replace x with 2. (2)2 4. 4 2. For radical expressions where the exponent of the variable inside the radical is even and the resulting simplified exponent is odd, you must use absolute value to ensure nonnegative results. x2 x . x3 xx . x4 x2 . x5 x2x . x6 x3 . Example 3 Simplify a Square Root with Variables Simplify 40x4y5 z3. 40x4y5 z3 23 5 x4 y5 z3 . In-Class Examples. Prime factorization. 22 2 5 x4 y4 y z2 z Product Property. 1 Simplify.. 2 2 5 x2 y2 y z z. a. 52 213 . Simplify. The absolute value of z ensures a nonnegative result.. 2x2y2z10yz . Power Point®. b. 72 62. Teaching Tip. Remind students that since 3 is a prime number, it cannot be factored further using prime factorization.. QUOTIENT PROPERTY OF SQUARE ROOTS You can divide square roots and simplify radical expressions that involve division by using the Quotient Property of Square Roots.. 2 Find 2 24 .. Quotient Property of Square Roots • Words. For any numbers a and b, where a 0 and b 0, the square root of a quotient is equal to the quotient of each square root.. • Symbols. a ba b. • Example. 43. 3 Simplify 45a4b5 c6. 3a 2b 2 |c 3 | 5b . 49 49 4 4. Study Tip Look Back To review the Quadratic Formula, see Lesson 10-4.. You can use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the quadratic equation ax2 bx c 0.. Interactive. ax2 bx c 0 Original equation b a. Chalkboard. c a. x2 x 0 Divide each side by a, a 0. (continued on the next page). www.algebra1.com/extra_examples. PowerPoint® Presentations. Lesson 11-1 Simplifying Radical Expressions 587. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools Lesson 11-1 Simplifying Radical Expressions 587.

(13) In-Class Example. 11y 33y b. 9 27 3 6 8. c. 4. c Subtract from each side.. b2. b a. a. b2. c a. 4a. 2. x 2ba . 2. 2a. 4a. 4ac b2 4a. b a. 2. 4a. b2 4a. Factor x2 x 2.. x 2a  4a b b 4ac x 2a 4a b2 4ac. b. 2. b b Complete the square; 2.. x2 x 2 2. Power Point®. 4 Simplify. 12 215 a. 5 5. c a. b a. x2 x . QUOTIENT PROPERTY OF SQUARE ROOTS. Take the square root of each side.. 2. 2. Study Tip. Remove the absolute value symbols and insert. 2. Plus or Minus Symbol. x . The symbol is used with the radical expression since both square roots lead to solutions.. x . b 2a. 2 4ac b 4a2 . Quotient Property of Square Roots. b 2a. b2 4 ac 2a. 4a2 2a . b b 4ac x 2. 2a. .. b Subtract from each side. 2a. Thus, we have derived the Quadratic Formula. A fraction containing radicals is in simplest form if no prime factors appear under the radical sign with an exponent greater than 1 and if no radicals are left in the denominator. Rationalizing the denominator of a radical expression is a method used to eliminate radicals from the denominator of a fraction.. Example 4 Rationalizing the Denominator Simplify.. 7x b. . 10 a. . 8. 3. 10 3 10 3 3 3. 3 . Multiply by . 30 . Product Property of Square Roots. 3. 3. 7x 7x 2 2 8 2. 2 7x . 2 . Multiply by . 14x . Product Property of Square Roots. 22 4. 2 c. . Prime factorization. 2. 2. 6. 2 6 2 6 6 6 12 . 6. . Product Property of Square Roots. 2 2 3 6. Prime factorization. 23 . 22 2. 3 . Divide the numerator and denominator by 2.. 6. 6. 3. 588. 6 . Multiply by . Chapter 11 Radical Expressions and Triangles. Differentiated Instruction Logical Have students use the same method that is shown for the derivation of the Quadratic Formula to solve actual quadratic equations. Have students turn back to Lesson 10-4 and solve an example problem using this method.. 588. Chapter 11 Radical Expressions and Triangles.

(14) Binomials of the form pq rs and pq rs are called conjugates.. For example, 3 2 and 3 2 are conjugates. Conjugates are useful when simplifying radical expressions because if p, q, r, and s are rational numbers, their product is always a rational number with no radicals. Use the pattern (a b)(a b) a2 b2 to find their product. 3 2 3 2 32 2 2 9 2 or 7. In-Class Example. Power Point®. 3 5 Simplify .. 15 32 23. 5 2 . a 3, b 2 . 3 Practice/Apply. 2 2 2 2 or 2. Example 5 Use Conjugates to Rationalize a Denominator 2 Simplify . 6 3 . 6 3 2 2 6 3 6 3 6 3 26 3 2 62 3. (a b)(a b) . 12 23 36 3. 3 2 3. 12 23 . Simplify.. 33. Study Notebook. 6 3 1 6 3. a2. . Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 11. • include examples of how to use the Product and Quotient Properties of Square Roots to simplify radical expressions. • include examples of how to rationalize denominators, with and without conjugates. • include any other item(s) that they find helpful in mastering the skills in this lesson.. b2. When simplifying radical expressions, check the following conditions to determine if the expression is in simplest form.. Simplest Radical Form A radical expression is in simplest form when the following three conditions have been met. 1. No radicands have perfect square factors other than 1. 2. No radicands contain fractions. 3. No radicals appear in the denominator of a fraction.. Answer. Concept Check. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4 5, 6, 13 7, 8 9, 10, 14 11, 12. 1 2 3 4 5. 1. Both x 4 and x 2 are positive even if x is a negative number.. x4 x2. See margin. 1. Explain why absolute value is not necessary for 1 1 a for a 0. a a 2. Show that a a a a a 3. OPEN ENDED Give an example of a binomial in the form ab cd and its conjugate. Then find their product. Sample answer: 22 33 and Simplify. 8. 2x2y315x . 22 33; 19. 4. 20 25. 5. 2 8 4. 7. 54a2b2 3ab6 . 8. 60x5y6. 10.. 3 10. 30 10. 6. 310 410 120. 8 83 2 11. 7 3 2. 4 6 2 9. 6 3 25 10 25 12. 2 4 8. Lesson 11-1 Simplifying Radical Expressions 589. Lesson 11-1 Simplifying Radical Expressions 589.

(15) Applications. About the Exercises … Organization by Objective • Product Property of Square Roots: 15–26, 39, 41 • Quotient Property of Square Roots: 27–38, 40 Odd/Even Assignments Exercises 15–40 are structured so that students practice the same concepts whether they are assigned odd or even problems.. Assignment Guide Basic: 15–41 odd, 42, 43, 50–53, 62–89 Average: 15–41 odd, 45–47, 50–53, 62–89 (optional: 54–61) Advanced: 16–40 even, 44–83 (optional: 84–89). 14. PHYSICS The period of a pendulum is the time required for it to make one complete swing back and forth. The formula of the period P of a pendulum is P 2. 51. A lot of formulas and calculations that are used in space exploration contain radical expressions. Answers should include the following. • To determine the escape velocity of a planet, you would need to know its mass and the radius. It would be very important to know the escape velocity of a planet before you landed on it so you would know if you had enough fuel and velocity to launch from it to get back into space. • The astronomical body with the smaller radius would have a greater escape velocity. As the radius decreases, the escape velocity increases.. ᐉ , where ᐉ is the length of the pendulum 32. in feet. If a pendulum in a clock tower is 8 feet long, find the period. Use 3.14 for . 3.14 s. ★ indicates increased difficulty. Practice and Apply Homework Help. Simplify. 25. 7x3y33y . 26. 6xy2z22xz . For Exercises. See Examples. 15. 18 32. 16. 24 26. 15–18, 41, 44–46 19–22, 39, 40, 48, 49 23–26 27–32, 42, 43, 47 33–38. 1. 18. 75 53. 19. 5 6. 30 . 20. 3 8 26. 2. 21. 730 26 845. 22. 23 527 90. 23. 40a4 2a210 . 3 4. 24. 50m3n5 5mn 22mn 147x6y7 25. . 5. 27.. Extra Practice See page 844.. x3xy 32. 3. Answer. 13. GEOMETRY A square has sides each measuring 27 feet. Determine the area of the square. 28 ft2. 2y  5 10 2 34. 2. 35. 27 22 26 23 36. 3. The speed of a roller coaster can be determined by evaluating a radical expression. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.. 16 123 37. 11 521 335 38. 10. 6 27 73 3 27 33 30. p p. 28. 31.. 2. 17. 80 45. 26. 72x3y4 z5. 310 35 64 10 5c c 5cd 4d 2d 5. 29.. 32. 2 6. 3. 18 54 92 33. 17 6 2. 25 34. . 2 36. 3 6. 4 37. 4 3 3. 4 8. 2t 4. 5y 9x 12x y . 2. 5. 8t. 10 35. . 7 2 37 ★ 38. 53 35. 39. GEOMETRY A rectangle has width 35 centimeters and length 4 10 centimeters. Find the area of the rectangle. 602 or 84.9 cm2. a. a. 40. GEOMETRY A rectangle has length meters and width meters. 8 2 What is the area of the rectangle? a 2 m 4 41. GEOMETRY The formula for the area A of a square with side length s is A s2. Solve this equation for s, and find the side length of a square having an area of 72 square inches. s A ; 62 in. PHYSICS For Exercises 42 and 43, use the following information. 1 2. The formula for the kinetic energy of a moving object is E mv2, where E is the kinetic energy in joules, m is the mass in kilograms, and v is the velocity in meters per second. 2E 42. Solve the equation for v. v m 43. Find the velocity of an object whose mass is 0.6 kilogram and whose kinetic energy is 54 joules. 6 5 or 13.4 m/s. . . 44. SPACE EXPLORATION Refer to the application at the beginning of the lesson. Find the escape velocity for the Moon in kilometers per second if 6.7. 1020 km s kg. G 2 , M 7.4. 1022 kg, and R 1.7. 103 km. How does this. compare to the escape velocity for Earth? 2.4 km/s; The Moon has a much lower 590. escape velocity than Earth.. Chapter 11 Radical Expressions and Triangles. Teacher to Teacher Judy Buchholtz. Dublin Scioto H.S., Dublin, OH. “To make algebra more meaningful to my students, I bring in the school police officer to discuss the formula used in Exercises 45–47.”. 590. Chapter 11 Radical Expressions and Triangles.

(16) INVESTIGATION For Exercises 45–47, use the following information.. NAME ______________________________________________ DATE. 2d 45. Write a simplified expression for the speed if f 0.6 for a wet asphalt road. 3 46. What is a simplified expression for the speed if f 0.8 for a dry asphalt road?. Example 1. Simplify 180 . Prime factorization of 180 Product Property of Square Roots Simplify. Simplify.. Simplify 120a2 b5 c4.. 120a2 b5 c4 3 3 2 b5 2 5 a c4 2 2 2 3 5 a2 b4 b c4 2 2 3 5 | a | b2 b c2 2 | a | b2c230b . GEOMETRY For Exercises 48 and 49, use the following information. Hero’s Formula can be used to calculate the area A of a triangle given the three side lengths a, b, and c.. Exercises Simplify.. 1. 28. 1 2. 2. 68 . 27 . s(s a )(s b )(s c), where s (a b c) A . 5. 162 . 48. Find the value of s if the side lengths of a triangle are 13, 10, and 7 feet. 15 49. Determine the area of the triangle. 203 or 34.6 ft2 50. CRITICAL THINKING. For any numbers a and b, where a 0 and b 0, ab a b .. Product Property of Square Roots. Example 2. 51.4 mph 46. 26d . Insurance investigators decide whether claims are covered by the customer’s policy, assess the amount of loss, and investigate the circumstances of a claim.. The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational square roots. When you simplify radical expressions with variables, use absolute value to ensure nonnegative results.. 180 2 2 33 5 22 32 5 2 3 5 65 . 47. An officer measures skid marks that are 110 feet long. Determine the speed of the car for both wet road conditions and for dry road conditions. 44.5 mph,. Insurance Investigator. p. 643 (shown) and p. 644 Simplifying Radical Expressions. Product Property of Square Roots. Lesson 11-1. Police officers can use the formula s 30fd to determine the speed s that a car was traveling in miles per hour by measuring the distance d in feet of its skid marks. In this formula, f is the coefficient of friction for the type and condition of the road.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 11-1 Study Guide and. 6. 3 6 . 7. 2 5 . 92 . 32 10. 9x4. 2 |a |. 4. 75 . 215 . 9. 4a2. 10 . 52 12. 128c6. 8 |c 3 | 2 . 10a 23 . 3x 2. 15. 20a2b4. 9 |x3|. 2415 . 53 8. 5 10 . 11. 300a4. 14. 3x2 3 3x4. 13. 410 36 . a 1 a 1 Simplify . a2 3a 1 a 1 a. 3. 60 . 217 . 16. 100x3y. 2 |a |b 25 . 10 | x |xy . 17. 24a4b2. 18. 81x4y2. 19. 150a2 b2c. 3c2 20. 72a6b. 5z8 21. 45x2y. 6z2 22. 98x4y. 2a 2 | b | 6 . 9x 2 | y |. 6 |a 3bc | 2b . 5 |ab | 6c . 3 |x |y 2z 45y . 7x 2 |y 3z | 2 . NAME ______________________________________________ DATE. Online Research For more information about a career as an insurance investigator, visit: www.algebra1.com/ careers Source: U.S. Department of Labor. 51. WRITING IN MATH. Skills Practice, 11-1 Practice (Average) Simplify.. How are radical expressions used in space exploration? Include the following in your answer: • an explanation of how you could determine the escape velocity for a planet and why you would need this information before you landed on the planet, and • a comparison of the escape velocity for two astronomical bodies with the same mass, but different radii.. 1. 24 26 . 2. 60 215 . 3. 108 63 . 4. 8 6 43 . 5. 7 14 72 . 6. 312 56 902 . 32a3. B. 48a3. C. 64a3. D. 96a3. 10. 108x6 y4z5 6| x 3 | y 2z 2 3z . 9. 50p5 5p 22p . 23 12. 8 6 . 4o5 2| m | n 2o 214o 11. 56m2n . 15. 19.. . 4 15 5 5. 1 7. 4. 2. 9ab 4ab4. 15 32 23. 3 21. . 2. 12 43 3. 8 22. 3 3 . 53 57 4. 921 37 37 1 27 26. 5 23. . 7 3 . 7 11. 18 x3. 4y 23y 3y2 3|y|. 5 2 . 5 32. 14.. 6k 17. 3k 8 . 3. 10 8 11 16. 11 32x 18. x 3b 20. 2b. 2 5 10 5. 13.. 52. If the cube has a surface area of 96a2, what is its volume? C A. 8. 27su3 3| u | 3su . 7. 43 318 366 . 3 4. Standardized Test Practice. ____________ PERIOD _____. p. 645 and Practice, 646Expressions (shown) Simplifyingp. Radical. Answer the question that was posed at the beginning of the lesson. See margin.. 24. . 25. SKY DIVING When a skydiver jumps from an airplane, the time t it takes to free fall a. 53. If x 81b2 and b. given distance can be estimated by the formula t . 0, then x B. A. 9b.. B. 9b.. C. 3b27 .. D. 27b3.. . 2s , where t is in seconds and s is in 9.8. meters. If Julie jumps from an airplane, how long will it take her to free fall 750 meters?. Surface area of a cube 6s2. about 12.4 s METEOROLOGY For Exercises 26 and 27, use the following information. To estimate how long a thunderstorm will last, meteorologists can use the formula t. . d3 , where t is the time in hours and d is the diameter of the storm in miles. 216. 26. A thunderstorm is 8 miles in diameter. Estimate how long the storm will last. Give your answer in simplified form and as a decimal. 83 . h 9. 1.5 h. 27. Will a thunderstorm twice this diameter last twice as long? Explain.. No; it will last about 4.4 h, or nearly 3 times as long. NAME ______________________________________________ DATE. WEATHER For Exercises 54 and 55, use the following information. The formula y 91.4 (91.4 t) 0.478 0.301x 0.02 can be used to find the windchill factor. In this formula, y represents the windchill factor, t represents the air temperature in degrees Fahrenheit, and x represents the wind speed in miles per hour. Suppose the air temperature is 12°.. Pre-Activity. 54. Use a graphing calculator to find the wind speed to the nearest mile per hour if it feels like 9° with the windchill factor. 7 mph. Reading the Lesson. ____________ PERIOD _____. Reading 11-1 Readingto to Learn Learn Mathematics Mathematics, p. 647 Simplifying Radical Expressions. ELL. How are radical expressions used in space exploration? Read the introduction to Lesson 11-1 at the top of page 586 in your textbook. Suppose you want to calculate the escape velocity for a spacecraft taking off from the planet Mars. When you substitute numbers in the formula, which number is sure to be the same as in the calculation for the escape velocity for a spacecraft taking off from Earth? the value of G. 1. a. How can you tell that the radical expression 28x2y4 is not in simplest form?. The radicand contains perfect square factors other than 1.. 55. What does it feel like to the nearest degree if the wind speed is 4 miles per hour? 6°F. www.algebra1.com/self_check_quiz. b. To simplify 28x2y4, you first find the You then apply the. Lesson 11-1 Simplifying Radical Expressions 591. prime factorization. Product Property of Square Roots. 4 7 x2 y4 is equal to the product 4 again to get a final answer of 2 | x | y27 .. of 28x2y4.. . In this case,. 7 x 2 y 4 . You can simplify. 2. Why is it correct to write y4 y2, with no absolute value sign, but not correct to write x2 x? NAME ______________________________________________ DATE. 11-1 Enrichment Enrichment,. ____________ PERIOD _____. p. 648. Squares and Square Roots From a Graph The graph of y x2 can be used to find the squares and square roots of numbers.. rationalizing the denominator. 4. What should you do to write the conjugate of a binomial of the form ab cd ? To write the conjugate of a binomial of the form ab cd ?. The arrows show that 32 9.. A small part of the graph at y x2 is shown below. A 1:10 ratio for unit length on the y-axis to unit length on the x-axis is used.. Example. Find 11 .. 3.3 The arrows show that 11 to the nearest tenth.. y. 12t 15 . 3. What method would you use to simplify ? y. To find the square of 3, locate 3 on the x-axis. Then find its corresponding value on the y-axis.. To find the square root of 4, first locate 4 on the y-axis. Then find its corresponding value on the x-axis. Following the arrows on the graph, you can see that 4 2.. Sample answer: The square of y 2 is y 4 and the expressions y 4 and y 2 both represent positive numbers for all values of y. Although it is true 2 2 that the square of x is x , when x is less than 0, x represents a positive quantity and x represents a negative quantity.. Change the plus sign to a minus sign; change the minus sign to a plus sign. O. x. Helping You Remember 5. What should you remember to check for when you want to determine if a radical expression is in simplest form?. Sample answer: Check radicands for perfect squares and fractions, and check fractions for radicals in the denominator.. 20. Lesson 11-1 Simplifying Radical Expressions 591. Lesson 11-1. Graphing Calculator.

(17) Extending the Lesson. 4 Assess. 1. x 2 x. Using the properties of exponents, simplify each expression. 1. 5. 57. x 2 x2. 1. 1. 56. x 2 x 2 x. Open-Ended Assessment Writing Have students copy the Concept Summary on p. 589 into their Study Notebooks. For each of the three conditions listed, have students provide an example of an expression that does not satisfy the condition and then show how they simplified the expression to satisfy the condition.. Radical expressions can be represented with fractional exponents. For example, 4. x2 x. 3. 58. x 2 or xx. a 5 a . a 6 or ★ 59. Simplify the expression 3 6. aa. a. 1 for y. ★ 60. Solve the equation y3 3 3. ★ 61. Write . 1. in simplest form.. s2t 2. 8. s5t4. 3 3. s18t 6. s . Maintain Your Skills Mixed Review. Find the next three terms in each geometric sequence. 62. 2, 6, 18, 54 162, 486, 1458 64. 384, 192, 96, 48 24, 12, 6 3 3 3 3 3 3 66. 3, , , , , 4 16 64 256 1024 4096. Getting Ready for Lesson 11-2. (Lesson 10-7). 63. 1, 2, 4, 8 16, 32, 64. 1 2 65. , , 4, 24 144, 864, 5184 9 3. 67. 50, 10, 2, 0.4 0.08, 0.016, 0.0032. 68. BIOLOGY A certain type of bacteria, if left alone, doubles its number every 2 hours. If there are 1000 bacteria at a certain point in time, how many bacteria will there be 24 hours later? (Lesson 10-6) 4,096,000. PREREQUISITE SKILL Students. will learn how to perform operations with radical expressions in Lesson 11-2. In order to perform operations with radical expressions, students must be able to use the Distributive Property to multiply binomials. Use Exercises 84–89 to determine your students’ familiarity with multiplying binomials.. 69. PHYSICS According to Newton’s Law of Cooling, the difference between the temperature of an object and its surroundings decreases in time exponentially. Suppose a cup of coffee is 95°C and it is in a room that is 20°C. The cooling of the coffee can be modeled by the equation y 75(0.875)t, where y is the temperature difference and t is the time in minutes. Find the temperature of the coffee after 15 minutes. (Lesson 10-6) 84.9°C Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 70. 6x2 7x 5 (3x 5)(2x 1). 71. 35x2 43x 12 (5x 4)(7x 3). 72. 5x2 3x 31 prime. 73. 3x2 6x 105 3(x 7)(x 5). 74.. 4x2. 12x 15 prime. 75. 8x2 10x 3 (4x 3)(2x 1). Find the solution set for each equation, given the replacement set. (Lesson 4-4) 76. y 3x 2; {(1, 5), (2, 6), (2, 2), (4, 10)} {(1, 5), (4, 10)} 77. 5x 2y 10; {(3, 5), (2, 0), (4, 2), (1, 2.5)} {(2, 0), (1, 2.5)} 78. 3a 2b 11; {(3, 10), (4, 1), (2, 2.5), (3, 2)} {(3, 10), (2, 2.5)} 3 1 1 79. 5 x 2y; (0, 1), (8, 2), 4, , (2, 1) 4, , (2, 1) 2 2 2. . . . Solve each equation. Then check your solution. (Lesson 3-3) 80. 40 5d 8. 81. 20.4 3.4y 6. h 82. 25 275 11. Getting Ready for the Next Lesson. PREREQUISITE SKILL Find each product. (To review multiplying binomials, see Lesson 8-7.) 84. (x 3)(x 2) x2 x 6 85.. 11t 6 88. (5x 3y)(3x y) 15x2 4xy 3y2 86. (2t 1)(t 6) 592. 592. Chapter 11 Radical Expressions and Triangles. r 29. 83. 65 1885. Chapter 11 Radical Expressions and Triangles. 2t 2. (a 2)(a 5) a2 7a 10. 87. (4x 3)(x 1) 4x2 x 3 89. (3a 2b)(4a 7b) 12a2 13ab 14b2.

(18) Operations with Radical Expressions. Lesson Notes. • Add and subtract radical expressions.. 1 Focus. • Multiply radical expressions.. can you use radical expressions to determine how far a person can see? The formula d . 3h represents 2. World’s Tall Structures. the distance d in miles that a person h feet high can see. To determine how much farther a person can see from atop the Sears Tower than from atop the Empire State Building, we can substitute the heights of both buildings into the equation.. 5-Minute Check Transparency 11-2 Use as a quiz or review of Lesson 11-1. Mathematical Background notes are available for this lesson on p. 584C.. 984 feet 1250 feet Eiffel Empire State Tower, Building, Paris New York. 1380 feet Jin Mau Building, Shanghai. 1450 feet 1483 feet Sears Petronas Tower, Towers, Chicago Kuala Lumpur. ADD AND SUBTRACT RADICAL EXPRESSIONS Radical expressions in which the radicands are alike can be added or subtracted in the same way that monomials are added or subtracted. Monomials. Radical Expressions. 2x 7x (2 7)x. 211 711 (2 7)11 911 . 9x. 152 32 (15 3)2 122. 15y 3y (15 3)y 12y. Notice that the Distributive Property was used to simplify each radical expression.. Example 1 Expressions with Like Radicands Simplify each expression. a. 43 63 53. 43 63 53 (4 6 5)3 53. Distributive Property Simplify.. b. 125 37 67 85 . Commutative. 125 37 67 85 125 85 37 67 Property (12 8)5 (3 6)7 Distributive Property 45 97. can you use radical expressions to determine how far a person can see? Ask students: • What is an assumption that the formula makes? Sample answer: That there are no obstructions that would block a person from seeing the entire distance. • Assuming that nothing blocks the view, how much farther can a person atop the Sears Tower see than a person atop the Empire State Building? about 3.3 miles farther • Mountain Climbing Mount Everest, the tallest mountain in the world, is about 12,000 feet above the plateau out of which it rises. How far would a climber be able to see from the peak of Mount Everest? about 134 miles. Simplify.. In Example 1b, 45 97 cannot be simplified further because the radicands are different. There are no common factors, and each radicand is in simplest form. If the radicals in a radical expression are not in simplest form, simplify them first. Lesson 11-2 Operations with Radical Expressions 593. Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 649–650 • Skills Practice, p. 651 • Practice, p. 652 • Reading to Learn Mathematics, p. 653 • Enrichment, p. 654 • Assessment, p. 699. Parent and Student Study Guide Workbook, p. 84 Prerequisite Skills Workbook, pp. 37–38. Transparencies 5-Minute Check Transparency 11-2 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 593.

(19) Example 2 Expressions with Unlike Radicands. 2 Teach. Simplify 220 345 180 .. ADD AND SUBTRACT RADICAL EXPRESSIONS. In-Class Examples. 220 22 5 3 32 5 62 5 345 180 2 2 22 5 3 32 5 62 5 . Power Point®. 225 335 65 . 1 Simplify each expression.. 45 95 65. a. 65 25 55 35. 195. b. 72 811 411 62 2 411 . The simplified form is 195.. 2 627 812 275 . You can use a calculator to verify that a simplified radical expression is equivalent to the original expression. Consider Example 2. First, find a decimal approximation for the original expression.. 443. KEYSTROKES:. MULTIPLY RADICAL EXPRESSIONS. In-Class Example. 2 2nd [2 ] 20 ) 180. ). ENTER. 3 2nd [2 ] 45 ). [2 ]. 2nd. 42.48529157. Next, find a decimal approximation for the simplified expression. Power Point®. KEYSTROKES:. 3 Find the area of a rectangle. 19 2nd [2 ] 5 ENTER. 42.48529157. Since the approximations are equal, the expressions are equivalent.. with a width of 46 210 and a length of 53 75. 1830 102. MULTIPLY RADICAL EXPRESSIONS Multiplying two radical expressions with different radicands is similar to multiplying binomials.. Example 3 Multiply Radical Expressions Find the area of the rectangle in simplest form. To find the area of the rectangle multiply the measures of the length and width.. 45 23 36 10. 45 23 36 10 First terms. Study Tip. Outer terms. Inner terms. Last terms. 45 36 45 10 23 36 23 10 . Look Back To review the FOIL method, see Lesson 8-7.. 1230 450 618 230 . Multiply.. 226 222 1230 45 3 30. Prime factorization. 1230 202 182 230 . Simplify.. 1430 382. Combine like terms.. The area of the rectangle is 1430 382 square units. 594 Chapter 11 Radical Expressions and Triangles. Differentiated Instruction Visual/Spatial Have students write all the perfect squares from 1 to 100 on an index card using a colored pen or pencil. Then have them rework Example 3. After they multiply the binomials, have students use their colored pen or pencil to circle the terms inside radicals that have perfect square factors. Using the same color, have students write the factorization in the line below each radical expression that can be factored. 594 Chapter 11 Radical Expressions and Triangles.

(20) Concept Check. 1. Explain why you should simplify each radical in a radical expression before adding or subtracting. to determine if there are any like radicands 2. Explain how you use the Distributive Property to simplify like radicands that are added or subtracted. See margin. 3. OPEN ENDED Choose values for x and y. Then find x y . 2. GUIDED PRACTICE KEY Exercises. Examples. 4, 5 6–9 10–13. 1 2 3. Sample answer: 2 3 2 26 3 or 5 26. Simplify each expression. 4. 43 73 113 . 5. 26 76 56 . 6. 55 320 5. 7. 23 12 43. 8. 35 56 320 95 56. 9. 83 3 9 93 3. Find each product. 10. 28 43 4 46 . Applications. 11. 4 5 3 5 17 75 . 12. GEOMETRY Find the perimeter and the area of a square whose sides measure 4 36 feet. P 16 126 ft; A 70 246 ft2 13. ELECTRICITY The voltage V required for a circuit is given by V PR , where P is the power in watts and R is the resistance in ohms. How many more volts are needed to light a 100-watt bulb than a 75-watt bulb if the resistance for both is 110 ohms? 10110 5330 14.05 volts. ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 14–21 22–29 30–48. 1 2 3. Simplify each expression. 14. 85 35 115 . 16. 215 615 315 715 . 15. 36 106 136 17. 519 619 1119 0. 18. 16x 2x 18x. 19. 35b 45b 115b 105b . Extra Practice. 20. 83 22 32 53. 21. 46 17 62 417 . 20. 133 2. 22. 18 12 8 52 23. 23. 6 23 12 . 24. 37 228 7. See page 844.. 25.. 6 43 250 332 22. 32 410 21. 46 62 ★ 26. 2 1 ★ 27. 10 25 2 2 5 5 17 537 ★ 28. 33 45 3 13 43 35 ★ 29. 6 74 328 10 17 . . . . . . About the Exercises … Organization by Objective • Add and Subtract Radical Expressions: 14–29 • Multiply Radical Expressions: 30–37 Odd/Even Assignments Exercises 14–39 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 42 requires the use of the Internet or other research materials.. Assignment Guide. 7. Find each product.. 34. 103 16. Study Notebook Have students— • include examples of how to add, subtract, and multiply radical expressions. • include any other item(s) that they find helpful in mastering the skills in this lesson.. 2. Guided Practice. 3 Practice/Apply. 30. 63 52 32 103. 31. 5210 32 102 310 . 32. 3 5 3 5 4. 33. 7 10 2 59 1410 . ★ 34. 6 8 24 2 . ★ 35. 5 2 14 35 37. ★ 36. 210 315 33 22 195. ★ 37. 52 35 210 3 152 115. Basic: 15–25 odd, 31, 33, 39, 41, 42, 49–76 Average: 15–39 odd, 41–44, 49–76 Advanced: 14–40 even, 45–70 (optional: 71–76). 38. GEOMETRY Find the perimeter of a rectangle whose length is. 87 45 inches and whose width is 27 35 inches. 20 7 25 in.. www.algebra1.com/extra_examples. Lesson 11-2 Operations with Radical Expressions 595. Answer 2. The Distributive Property allows you to add like terms. Radicals with like radicands can be added or subtracted.. Lesson 11-2 Operations with Radical Expressions 595.

(21) NAME ______________________________________________ DATE. 39. GEOMETRY The perimeter of a rectangle is 23 411 6 centimeters,. ____________ PERIOD _____. Study Guide andIntervention Intervention, 11-2 Study Guide and p. 649 (shown) and p. 650 Operations with Radical Expressions. and its length is 211 1 centimeters. Find the width.. Add and Subtract Radical Expressions. When adding or subtracting radical expressions, use the Associative and Distributive Properties to simplify the expressions. If radical expressions are not in simplest form, simplify them.. Example 1. Simplify 106 53 63 46 .. 53 63 46 (10 4)6 (5 6)3 106 6 6 3 . Example 2. Simplify.. 1 2. Simplify.. Lesson 11-2. Distributive Property. 1. 25 45 65 . 2. 6 46 36 . 2 5 . 6. 23 6 53 33 . 7. 12 23 53 3 . 12. 12 . 1 176 6 6. . 14. 80 20 180 85 . . 1 5. 42. A person atop the Empire State Building can see approximately 4.57 miles farther than a person atop the Texas Commerce Tower in Houston. Explain how you could find the height of the Texas Commerce Tower. See margin.. 1 73 3 3. 15. 50 18 75 27 8 2 2 3 16. 23 445 2 17. 125 2. 41. How much farther can a person see from atop the Sears Tower than from atop the Empire State Building? 587 253 3.34 mi. 10. 54 24 56 . 1 43 3 3. . 6 . 8. 3 6 3 2 50 24 5 6 2 2. 9. 8a 2a 52a 62a . 13. 54 . DISTANCE For Exercises 41 and 42, refer to the application at the beginning of the lesson.. 4. 375 25 153 25 . 5. 20 25 35 . 1 235 3 5. 3 3. 18.. . . 2 33 4 3. 1 83 3 3. 1 6 12 3. 125 . 283 12. 1. 830 430 430 . 2. 25 75 55 45 . 3. 713x 1413x 213x 513x . 4. 245 420 145 . 5. 40 10 90 410 . 6. 232 350 318 142 . 7. 27 18 300 32 133 . 8. 58 320 32 62 65 . Source: www.the-skydeck.com. NAME ______________________________________________ DATE. ____________ PERIOD _____. p. 651 and Practice, (shown) Operationsp. with652 Radical Expressions. Simplify each expression.. 9. 14 . . 2 614 7 7. 10. 50 32 . 11. 519 428 819 63 . . Online Research Data Update What are the tallest buildings and towers in the world today? Visit www.algebra1.com/data_update to learn more.. Distance The Sears Tower was the tallest building in the world from 1974 to 1996. You can see four states from its roof: Michigan, Indiana, Illinois, and Wisconsin.. Skills Practice, 11-2 Practice (Average). ENGINEERING For Exercises 43 and 44, use the following information. rate F of water passing through it in gallons per minute.. 43. Find the radius of a pipe that can carry 500 gallons of water per minute. Round to the nearest whole number. 6 in.. 1 172 2 2. 10 33 . 44. An engineer determines that a drainpipe must be able to carry 1000 gallons of water per minute and instructs the builder to use an 8-inch radius pipe. Can the builder use two 4-inch radius pipes instead? Justify your answer.. Find each product.. (10 15 ) 215 310 13. 6. 14. 5 (52 48 ) 310 . 15. 27 (312 58 ) 1221 2014 16. (5 15 ) 40 1015 2. 17. (10 6 )(30 18 ) 43 . 18. (8 12 )(48 18 ) 36 146 . 19. (2 28 )(36 5 ). 20. (43 25 )(310 56 ). 303 510 . F relates the radius r of a drainpipe in inches to the flow 5. The equation r . 12. 310 75 240 412 . 319 117 . 54 cm. the diagonals of the rhombus. What is the area of the rhombus at the right? 156 cm2. Simplify.. Exercises. 3. 8 2 . 36 cm. A d1d2, where d1 and d2 are the lengths of. Simplify.. Simplify each expression.. . 40. GEOMETRY A formula for the area A of a rhombus can be found using the formula. Associative and Distributive Properties. Simplify 312 575 .. 575 3 22 3 5 52 3 312 3 23 5 53 6 3 253 31 3. 11. 3 . 3 2 cm. No, each pipe would need to carry 500 gallons per minute, so the pipes would need at least a 6-inch radius.. 230 302 . MOTION For Exercises 45–47, use the following information. The velocity of an object dropped from a certain height can be found using the. SOUND For Exercises 21 and 22, use the following information. t 27 3, The speed of sound V in meters per second near Earth’s surface is given by V 20 where t is the surface temperature in degrees Celsius.. formula v 2gd , where v is the velocity in feet per second, g is the acceleration due to gravity, and d is the distance in feet the object drops.. 21. What is the speed of sound near Earth’s surface at 15°C and at 2°C in simplest form?. 2402 m/s, 10011 m/s 22. How much faster is the speed of sound at 15°C than at 2°C?. 2402 10011 . 45. Find the speed of an object that has fallen 25 feet and the speed of an object that has fallen 100 feet. Use 32 feet per second squared for g. 40 ft/s; 80 ft/s. 7.75 m/s. GEOMETRY For Exercises 23 and 24, use the following information. A rectangle is 57 23 centimeters long and 67 33 centimeters wide.. 23 cm 23. Find the perimeter of the rectangle in simplest form. 227. 46. When you increased the distance by 4 times, what happened to the velocity?. 24. Find the area of the rectangle in simplest form. 192 321 cm2. NAME ______________________________________________ DATE. The velocity doubled. 47. MAKE A CONJECTURE Estimate the velocity of an object that has fallen 225 feet. Then use the formula to verify your answer. The velocity should be. ____________ PERIOD _____. Reading 11-2 Readingto to Learn Learn MathematicsELL Mathematics, p. 653 Operations with Radical Expressions Pre-Activity. 9 or 3 times the velocity of an object falling 25 feet; 3 40 120 ft/s, 25) 120 ft/s. 2(32)(2. How can you use radical expressions to determine how far a person can see? Read the introduction to Lesson 11-2 at the top of page 593 in your textbook.. 48. WATER SUPPLY The relationship between a city’s size and its capacity to supply. Suppose you substitute the heights of the Sears Tower and the Empire State Building into the formula to find how far you can see from atop each building. What operation should you then use to determine how much farther you can see from the Sears Tower than from the Empire State Building?. water to its citizens can be described by the expression 1020P 1 0.01P , where P is the population in thousands and the result is the number of gallons per minute required. If a city has a population of 55,000 people, how many gallons per minute must the city’s pumping station be able to supply?. subtraction. Reading the Lesson. about 7003.5 gal/min. 1. Indicate whether the following expressions are in simplest form. Explain your answer.. 596 Chapter 11 Radical Expressions and Triangles. a. 63 12 . No; 12 can be simplified to 22 3 or 23 . b. 126 710 . Yes; both the addends are radical expressions in simplest form, the radicands are different, and there are no common factors. 2. Below the words First terms, Outer terms, Inner terms, and Last terms, write the products you would use to simplify the expression (215 315 )(63 52 ). First terms. Outer terms. Inner terms. (215 )(63 ). (215 )(52 ) . (35 )(63). Last terms . (35 )(52 ). Helping You Remember 3. How can you use what you know about adding and subtracting monomials to help you remember how to add and subtract radical expressions?. Sample answer: Check that the addends have been simplified. Next, group addends that involve like radicals. Then use the Distributive Property to combine the addends that involve like radicals.. 596. NAME ______________________________________________ DATE. 11-2 Enrichment Enrichment,. ____________ PERIOD _____. p. 654. 42. Approximately 1000 feet; determine h so that. The Wheel of Theodorus The Greek mathematicians were intrigued by problems of representing different numbers and expressions using geometric constructions.. 1. D. C 1. Theodorus, a Greek philosopher who lived about 425 B.C., is said to have discovered a way to construct the sequence 1, 2, 3, 4, … .. B 1. The beginning of his construction is shown. You start with an isosceles right triangle with sides 1 unit long. O. 1. Use the figure above. Write each length as a radical expression in simplest form. 1. line segment AO. 1 . 2. line segment BO. 2 . 3. line segment CO. 3 . 4. line segment DO. 4 . Chapter 11 Radical Expressions and Triangles. Answer. A. 3(1250) 2. 3h 4.57; 2. may use guess and test, graphical, or analytical methods..

(22) 49. CRITICAL THINKING Find a counterexample to disprove the following statement. Sample answer: a 4, b 9: 4 9 4 9 For any numbers a and b, where a. 4 Assess. 0, a b a b.. 0 and b. a b a b 50. CRITICAL THINKING Under what conditions is true? a 0 or b 0 or both 2. 51. WRITING IN MATH. 2. 2. Answer the question that was posed at the beginning of the lesson. See margin.. How can you use radical expressions to determine how far a person can see? Include the following in your answer: • an explanation of how this information could help determine how far apart lifeguard towers should be on a beach, and • an example of a real-life situation where a lookout position is placed at a high point above the ground.. Standardized Test Practice. C. 7 57. B. 47. D. 77. PREREQUISITE SKILL Students will learn how to solve radical equations in Lesson 11-3. In order to solve radical equations, students will need to be able to recognize and find special products. Use Exercises 71–76 to determine your students’ familiarity with finding special products.. 53. Simplify 34 12 . D 2. A. 43 6. B. 283. C. 28 163. D. 48 283. Maintain Your Skills Mixed Review. 56. 14xy y. Simplify.. (Lesson 11-1). 54. 40 210 . 55. 128 82. 5 50 57. 8 2. 58.. . Find the nth term of each geometric sequence. 60. a1 4, n 6, r 4. 2 y3 56. 196x . 225c4d 5c2d 18c2 2. 59.. 5103. 4096. Assessment Options. 314 63a 128a3b2 16ab. . Quiz (Lessons 11-1 and 11-2) is available on p. 699 of the Chapter 11 Resource Masters.. (Lesson 10-7). 61. a1 7, n 4, r 9. 62. a1 2, n 8, r 0.8. 0.4194304. Solve each equation by factoring. Check your solutions. (Lesson 9-5) 6 9 36 63. 81 49y2 64. q2 0 121 11 7 5 5 65. 48n3 75n 0 , 0, 66. 5x3 80x 240 15x2 {4, 3, 4} 4 4. . . . . . Solve each inequality. Then check your solution. (Lesson 6-2) 5 w 7k 67. 8n 5 n 68. 14 w 126 69. 9 2 8. . 21 10. k. 3 5. 70. PROBABILITY A student rolls a die three times. What is the probability that each roll is a 1? (Lesson 2-6) 1 216. Getting Ready for the Next Lesson. PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.) 71. (x 2)2 x2 4x 4. 72. (x 5)2 x2 10x 25 73. (x 6)2 x2 12x 36. 74. (3x 1)2 9x2 6x 1 75. (2x 3)2. www.algebra1.com/self_check_quiz. 4x2 12x 9. Speaking Ask students to compare and contrast adding, subtracting, and multiplying radical expressions and variable expressions. What are the similarities? What are the differences? Allow students to give examples on the chalkboard if they wish.. Getting Ready for Lesson 11-3. 52. Find the difference of 97 and 228 . C A. Open-Ended Assessment. 76. (4x 7)2. 16x2 56x 49. Lesson 11-2 Operations with Radical Expressions 597. Answer 51. The distance a person can see is related to the height of the person using d . 3h . Answers should 2. include the following. • You can find how far each lifeguard can see from the height of the lifeguard tower. Each tower should have some overlap to cover the entire beach area. • On early ships, a lookout position (Crow’s nest) was situated high on the foremast. Sailors could see farther from this position than from the ship’s deck.. Lesson 11-2 Operations with Radical Expressions 597.

(23) Lesson Notes. Radical Equations • Solve radical equations.. 1 Focus. • Solve radical equations with extraneous solutions.. 5-Minute Check Transparency 11-3 Use as a quiz or review of Lesson 11-2.. Vocabulary • radical equation • extraneous solution. Mathematical Background notes are available for this lesson on p. 584D. are radical equations used to find freefall times? Ask students: h 4. are radical equations used to find free-fall times? Skydivers fall 1050 to 1480 feet every 5 seconds, reaching speeds of 120 to 150 miles per hour at terminal velocity. It is the highest speed they can reach and occurs when the air resistance equals the force of gravity. With no air resistance, the time t in seconds that it takes an object to fall h h feet can be determined by the equation t . 4 How would you find the value of h if you are given the value of t?. h that contain radicals with RADICAL EQUATIONS Equations like t 4. • Is in simplest form?. variables in the radicand are called radical equations. To solve these equations, first isolate the radical on one side of the equation. Then square each side of the equation to eliminate the radical.. Explain. Yes, since there is no radical in the denominator, the expression is in simplest form. x 4. • How would you solve t . Example 1 Radical Equation with a Variable. for x? Multiply each side by 4. • What do you think might be a way to remove the radical sign from 4t h? Square each side of the equation.. FREE-FALL HEIGHT Two objects are dropped simultaneously. The first object reaches the ground in 2.5 seconds, and the second object reaches the ground 1.5 seconds later. From what heights were the two objects dropped? Find the height of the first object. Replace t with 2.5 seconds. h t . Original equation. 4. h 2.5 . Replace t with 2.5.. 4. 10 h. Multiply each side by 4.. 102 h 100 h. 2. Square each side. Simplify.. h CHECK t 4. 100 t 4 10 t 4. t 2.5. Original equation h 100 100 10 Simplify.. The first object was dropped from 100 feet. 598. Chapter 11 Radical Expressions and Triangles. Resource Manager Workbook and Reproducible Masters Chapter 11 Resource Masters • Study Guide and Intervention, pp. 655–656 • Skills Practice, p. 657 • Practice, p. 658 • Reading to Learn Mathematics, p. 659 • Enrichment, p. 660. Graphing Calculator and Spreadsheet Masters, p. 43 Parent and Student Study Guide Workbook, p. 85. Transparencies 5-Minute Check Transparency 11-3 Answer Key Transparencies. Technology Interactive Chalkboard.

No results found