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(1)Notes Introduction In this unit, students will be introduced to nonlinear functions. Students will first learn about polynomials and operations involving monomials and polynomials. Students then learn various methods of factoring, and are finally introduced to quadratic and exponential functions.. Not all real-world situations can be modeled using a linear function. In this unit, you will learn about polynomials and nonlinear functions.. Polynomials Polynomials and Nonlinear Nonlinear Functions. Assessment Options Unit 3 Test Pages 641–642 of the Chapter 10 Resource Masters may be used as a test or review for Unit 3. This assessment contains both multiple-choice and short answer items.. Chapter 8 Polynomials. Chapter 9 Factoring. TestCheck and Worksheet Builder. Chapter 10 Quadratic and Exponential Functions. This CD-ROM can be used to create additional unit tests and review worksheets.. 406 Unit 3. 406. Unit 3 Polynomials and Nonlinear Functions. Polynomials and Nonlinear Functions.

(2) Teaching Suggestions Have students study the USA TODAY Snapshot. • Ask students what fraction of people believe that evidence will be discovered that life exists in this or other 2 galaxies. 3. • Take an informal poll in the class and compare the results to the USA TODAY results. • Point out to students that in their WebQuest they will be designing a display about the planets in the solar system. Additional USA TODAY Snapshots appearing in Unit 3: Chapter 8 A sweet holiday season (p. 427) Chapter 9 Number of domain registrations climbs (p. 494) Chapter 10 Spending more on eating out (p. 561) Making gains (p. 563) Grand Canyon Visitors (p. 564). Pluto Is Falling From Status as Distant Planet Source: USA TODAY, March 28, 2001. “Like any former third-grader, Catherine Beyhl knows that the solar system has nine planets, and she knows a phrase to help remember their order: ‘My Very Educated Mother Just Served Us Nine Pizzas.’ But she recently visited the American Museum of Natural History’s glittering new astronomy hall at the Hayden Planetarium and found only eight scale models of the planets. No Pizza—no Pluto.” In this project, you will examine how scientific notation, factors, and graphs are useful in presenting information about the planets. Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 3.. USA TODAY Snapshots® Are we alone in the universe? Adults who believe that during the next century evidence will be discovered that shows:. Life exists only on Earth. 28%. Other life in this or other galaxies. 66%. 6%. Don’t know. Source: The Gallup Organization for the John Templeton Foundation By Cindy Hall and Sam Ward, USA TODAY. Lesson Page. 8-3 429. 9-1 479. 10-2 537 Unit 3. Polynomials and Nonlinear Functions. 407. 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 10, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 3 Polynomials and Nonlinear Functions 407.

(3) Polynomials Chapter Overview and Pacing. PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. 2 2 (with 8-1 (with 8-1 Follow-Up) Follow-Up). Multiplying Monomials (pp. 410–416) • Multiply monomials. • Simplify expressions involving powers of monomials. Follow-Up: Use paper prisms to investigate surface area and volume.. Basic/ Average. Advanced. 0.5. 0.5. Dividing Monomials (pp. 417–423) • Simplify expressions involving the quotient of monomials. • Simplify expressions containing negative exponents.. 2. 2. Scientific Notation (pp. 425–430) • Express numbers in scientific notation and standard notation. • Find products and quotients of numbers expressed in scientific notation.. 1. 1. 0.5. 0.5. 2 (with 8-4 Preview). 1. 1 (with 8-4 Preview). 0.5. Adding and Subtracting Polynomials (pp. 437–443) Preview: Use algebra tiles to add and subtract polynomials. • Add polynomials. • Subtract polynomials.. 2. 2. 1. 1. Multiplying a Polynomial by a Monomial (pp. 444–449) • Find the product of a monomial and a polynomial. • Solve equations involving polynomials.. 2. 1. 1. 0.5. 2 (with 8-7 Preview). 2. 1 (with 8-7 Preview). 1. Special Products (pp. 458–463) • Find squares of sums and differences. • Find the product of a sum and a difference.. 1. 1. 0.5. 0.5. Study Guide and Practice Test (pp. 464–469) Standardized Test Practice (pp. 470–471). 1. 1. 0.5. 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 16. 14. 8. 7. Polynomials (pp. 431–436) Preview: Use algebra tiles to model polynomials. • Find the degree of a polynomial. • Arrange the terms of a polynomial in ascending or descending order.. Multiplying Polynomials (pp. 450–457) Preview: Use algebra tiles to find the product of two binomials. • Multiply two binomials by using the FOIL method. • Multiply two polynomials by using the Distributive Property.. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 408A Chapter 8 Polynomials. 1.5 1.5 (with 8-1 (with 8-1 Follow-Up) Follow-Up).

(4) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 455–456. 457–458. 459. 460. 461–462. 463–464. 465. 466. 467–468. 469–470. 471. 472. 473–474. 475–476. 477. 478. 479–480. 481–482. 483. 484. 485–486. 487–488. 489. 490. 491–492. 493–494. 495. 496. 497–498. 499–500. 501. 502. 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 8 RESOURCE MASTERS. See pages T12–T13.. Materials. 59. 8-1. 8-1. 18. (Follow-Up: centimeter grid paper). 60. 8-2. 8-2. 19. graphing calculator. SC 15, SM 67–70. 61. 8-3. 8-3. 20. graphing calculator. GCS 37. 62. 8-4. 8-4. (Preview: algebra tiles). 63. 8-5. 8-5. (Preview: algebra tiles). SC 16. 64. 8-6. 8-6. 21. GCS 38. 65. 8-7. 8-7. 22. 518. 66. 8-8. 8-8. 23. 503–516, 520–522. 67. 517. 33–36. 517, 519. 518. (Preview: algebra tiles, product mat). *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 8 Polynomials 408B.

(5) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 1, students apply the order of operations, and the Distributive, Commutative, and Associative Properties to simplify expressions. Students perform operations with real numbers in Chapter 2.. This Chapter This chapter helps students master operations with monomials and polynomials. It begins with connecting the multiplication and division of monomials with variables and exponents to multiplying and dividing real numbers. Students convert numbers from standard notation to scientific notation, and vice versa, then multiply and divide numbers written in scientific notation. Polynomials are defined, and students learn to perform operations with them. Finally, students learn patterns for finding products of some special polynomials.. Multiplying Monomials A monomial is a number, variable, or a product of a number and one or more variables. Monomials do not involve the addition, subtraction, or division of variables. Monomials that only involve real numbers are called constants. This is because their value does not change. When multiplying monomials, use the Commutative and Associative Properties to group constants together and group powers with the same base together. To multiply powers with the same base, add the exponents. Multiply the exponents to find a power of a power. When finding the power of a product, find the power of each factor. A monomial expression is simplified when each base appears exactly once, there are no powers of powers, and all fractions are in simplest form.. Dividing Monomials Exponents are subtracted when dividing two powers that have the same base. To find the power of a quotient, find the power of both the numerator and the denominator. If the numerator and denominator of a fraction are the same, the value of the fraction is 1. Therefore, if both the numerator and denominator have the same base raised to the same exponent, the value of the fraction is 1. Using the Quotient of Powers Property, the exponents of the original fraction are subtracted and the fraction simplifies to the base raised to the zero power. So it follows that any base raised to the zero power equals one. A positive number raised to a negative power represents the reciprocal of the number with the opposite or positive exponent. Likewise, if a negative exponent appears in the denominator of a fraction, this power is equivalent to a fraction with this same base raised to the opposite power in its numerator.. Scientific Notation. Future Connections Students will apply their knowledge of multiplying and dividing monomials and polynomials to factoring polynomials in Chapter 9. The concepts are essential to simplifying and solving many problems involving upper level mathematics and science.. 408C. Chapter 8 Polynomials. When a number is written in scientific notation, the power of 10 is the number of places the decimal moves. To convert a number that is in scientific notation to one in standard form, move the decimal the number of places indicated by the exponent. A positive exponent is used to represent a number that is greater than or equal to 10 in standard form. A negative exponent represents a number that is less than 1 in standard form. To translate a number in standard form to scientific notation, first move the decimal to the right of the first non-zero digit. Then write the appropriate power of 10 to the right of the number. Keep in mind what the sign of the power indicates. Numbers in scientific notation can be multiplied or divided. First multiply or divide the decimals. Then apply.

(6) either the Product of Powers or the Quotient of Powers Property to simplify the powers of 10. Next, rewrite the decimal in scientific notation and simplify the powers of 10. The result is in scientific notation, but it can also be presented in standard form if preferred.. Polynomials A polynomial is a monomial, or a sum of monomials. Remember that subtraction can be rewritten as addition. A monomial has only one term and is considered a type of polynomial. Some polynomials that contain more than one monomial, or term, have special names. A binomial has two terms and a trinomial has three terms. All others are just referred to as polynomials. The degree of a monomial is the sum of the exponents of all its variables. The term with the greatest degree determines the degree of a polynomial. Usually the terms of a polynomial are arranged so that the powers of one variable are in ascending (increasing) or descending (decreasing) order. This aids in reading and understanding the polynomial.. Adding and Subtracting Polynomials To add polynomials, combine like terms. Like terms have the same variable bases with the same exponents. The coefficients of like terms are added using the rules for adding real numbers. The bases and exponents of these terms, however, remain the same. The rule for subtracting polynomials is the same as the rule for subtracting integers. First replace each term of the second polynomial with its additive inverse or opposite. Then combine like terms using the rules for adding real numbers.. Multiplying a Polynomial by a Monomial To multiply a polynomial by a monomial, apply the Distributive Property by multiplying each term in the polynomial by the monomial. Use the rules for multiplying monomials. If the monomial is negative, don’t forget to apply the rules for multiplying real numbers. Be sure to simplify by combining any like terms. Equations may contain polynomials. To solve these equations, first simplify each side using the order of operations by multiplying, adding, and subtracting as indicated. Then apply the rules for solving multi-step equations and equations with variables on both sides.. Multiplying Polynomials The Distributive Property is applied twice when multiplying two binomials. Multiply the first term of the first binomial by each term of the second binomial. Do the same with the second term of the first binomial. Then combine like terms. This results in a multiplying pattern called the FOIL method. You multiply the First terms, Outer terms, Inside terms, and Last terms of the binomials. The Distributive Property is used to multiply any two polynomials. The product is not in simplest terms until all like terms have been combined.. Special Products The Distributive Property and FOIL method can always be used to multiply polynomials. However, some binomial products have patterns that make their multiplication simpler. One product involves the multiplying of two identical binomials, called the square of a sum. The pattern is (a b)(a b) a2 2ab b2. There is also a pattern for the square of a difference: (a b)(a b) a2 2ab b2. Note that the only difference in the two patterns is the sign before the middle term. A third pattern exists for the product of a sum and a difference: (a b)(a b) a2 b2. The products of the Outer terms and the Inner terms add to zero. While it is not essential to learn these patterns, identifying when to use them can make simplifying these products quicker and less laborious.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Multiplying Monomials (Lesson 22) • Dividing Monomials (Lesson 23) • Adding and Subtracting Polynomials (Lesson 24) • Multiplying a Polynomial by a Monomial (Lesson 25) • Multiplying Polynomials (Lesson 26) Chapter 8 Polynomials 408D.

(7) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 409, 415, 423, 430, 436, 443, 449, 457 Practice Quiz 1, p. 430 Practice Quiz 2, p. 449. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 33–36 Quizzes, CRM pp. 517–518 Mid-Chapter Test, CRM p. 519 Study Guide and Intervention, CRM pp. 455–456, 461–462, 467–468, 473–474, 479–480, 485–486, 491–492, 497–498. Mixed Review. pp. 415, 423, 430, 436, 443, 449, 457, 463. Cumulative Review, CRM p. 520. Error Analysis. Find the Error, pp. 413, 421, 441 Common Misconceptions, pp. 420, 432, 454. Find the Error, TWE pp. 413, 421 Unlocking Misconceptions, TWE pp. 421, 433 Tips for New Teachers, TWE pp. 426, 459. Standardized Test Practice. pp. 415, 420, 421, 423, 430, 436, 443, 448, 457, 463, 469, 470–471. TWE pp. 470–471 Standardized Test Practice, CRM pp. 521–522. Open-Ended Assessment. Writing in Math, pp. 415, 423, 430, 436, 443, 448, 457, 463 Open Ended, pp. 413, 421, 428, 434, 441, 446, 455, 461 Standardized Test, p. 471. Modeling: TWE pp. 415, 436, 449 Speaking: TWE pp. 423, 457 Writing: TWE pp. 430, 443, 463 Open-Ended Assessment, CRM p. 515. Chapter Assessment. Study Guide, pp. 464–468 Practice Test, p. 469. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 503–508 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 509–514 Vocabulary Test/Review, CRM p. 516. 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.. 408E. Chapter 8 Polynomials.

(8) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 8-1. 18 Performing Operations with Exponents II. 8-2. 19 Performing Operations with Exponents I. 8-3. 20 Using Scientific Notation. 8-6. 21 Simplifying Polynomial Expressions. 8-7. 22 Multiplying Binomial Expressions I. 8-8. 23 Multiplying Binomial Expressions II. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. 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. 59–67 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.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 409 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 413, 421, 428, 434, 441, 446, 455, 461) • Reading Mathematics, p. 424 • Writing in Math questions in every lesson, pp. 415, 423, 430, 436, 443, 448, 457, 463 • Reading Study Tip, pp. 410, 425 • WebQuest, p. 429 Teacher Wraparound Edition • Foldables Study Organizer, pp. 409, 464 • Study Notebook suggestions, pp. 413, 416, 421, 424, 428, 431, 434, 438, 446, 451, 455, 461 • Modeling activities, pp. 415, 436, 449 • Speaking activities, pp. 423, 457 • Writing activities, pp. 430, 443, 463 • Differentiated Instruction, (Verbal/Linguistic), p. 460 • ELL Resources, pp. 408, 414, 422, 424, 429, 435, 442, 448, 456, 460, 462, 464 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 8 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 8 Resource Masters, pp. 459, 465, 471, 477, 483, 489, 495, 501) • 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 2.1, 3.1–3.4, 6.2, 6.3, 7.5, 7.7, 7.8 For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 8 Polynomials 408F.

(9) Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. 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.. Polynomials • Lessons 8-1 and 8-2 Find products and quotients of monomials. • Lesson 8-3 Express numbers in scientific and standard notation. • Lesson 8-4 Find the degree of a polynomial and arrange the terms in order. • Lessons 8-5 through 8-7 Add, subtract, and multiply polynomial expressions. • Lesson 8-8 Find special products of binomials.. Key Vocabulary • • • • •. monomial (p. 410) scientific notation (p. 425) polynomial (p. 432) binomial (p. 432) FOIL method (p. 453). Operations with polynomials, including addition, subtraction, and multiplication, form the foundation for solving equations that involve polynomials. In addition, polynomials are used to model many real-world situations. In Lesson 8-6, you will learn how to find the distance that runners on a curved track should be staggered.. Lesson 8-1 8-1 Follow-Up 8-2 8-3 8-4 Preview 8-4 8-5 Preview 8-5 8-6 8-7 Preview 8-7 8-8. NCTM Standards. Local Objectives. 2, 6, 8, 9, 10 2, 3, 6, 7 2, 6, 8, 9, 10 1, 6, 8, 9, 10 2, 10 2, 6, 8, 9, 10 2, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 10 2, 6, 8, 9, 10 2, 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 408. Chapter 8 Polynomials. 408 Chapter 8 Polynomials. 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 8 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 8 test..

(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 8. For Lessons 8-1 and 8-2. Exponential Notation. Write each expression using exponents. (For review, see Lesson 1-1.) 1. 2 2 2 2 2 25. 2. 3 3 3 3 34. 3. 5 5 52. 4. x x x x 3. . 1 1 1 1 1 1 5 a a c c c 5. a a a a a a a6 6. x x y y y x 2y 3 7. 8. 2 2 2 2 2 2 b b d d d. ab2dc 3. For Lessons 8-1 and 8-2. Evaluating Powers. Evaluate each expression. (For review, see Lesson 1-1.) 9. 32 9 13.. (6)2. 10. 43 64. 36. 14.. (3)3. 27. 11. 52 25 2 4 16 15. 3 81. 12. 104 10,000 7 2 49 16. 8 64. For Lessons 8-1, 8-2, and 8-5 through 8-8. Area and Volume. Find the area or volume of each figure shown below. (For review, see pages 813–817.) 17.. 63 yd2 18.. 19. 6m. 9 yd. 36 m2 or about 113.04 m2. 20.. 125 cm3. 4 ft. 5 cm 3 ft 7 ft. 84 ft3. 5 cm. 5 cm. This section provides a review of the basic concepts needed before beginning Chapter 8. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 33–36. 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. For Lesson. Prerequisite Skill. 8-2 8-3 8-4 8-5 8-6 8-7 8-8. Simplifying Fractions (p. 415) Products of Powers (p. 423) Evaluating Expressions (p. 430) Simplifying Expressions (p. 436) Distributive Property (p. 443) Products of Powers (p. 449) Power of a Power, Power of a Product (p. 457). Make this Foldable to help you organize information about polynomials. Begin with a sheet of 11" by 17" paper. Fold. Open and Fold Fold a 2" tab along the width. Then fold the rest in fourths.. Fold in thirds lengthwise.. Poly. Mon.. Label. Draw lines along folds and label as shown.. Reading and Writing. As you read and study the chapter, write examples and notes. for each operation.. Chapter 8 Polynomials 409. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data using a Table to Make Comparisons After students make their Foldable tables, have them label the columns and rows as illustrated. Students use their Foldables to take notes and write examples for each operation. Have students compare different functions. For example, compare adding monomials and adding polynomials. Remind students that comparing involves determining a trait to be compared and then finding the similarities and differences in that trait. Chapter 8 Polynomials 409.

(11) Lesson Notes. • Simplify expressions involving powers of monomials.. Vocabulary • monomial • constant. Mathematical Background notes are available for this lesson on p. 408C.. The table shows the braking distance for a vehicle at certain speeds. If s represents the speed in miles per hour, then the approximate number of feet that the driver must 1 apply the brakes is s2. Notice that 20. when speed is doubled, the braking distance is quadrupled.. 20. 20. 30 45 40. 80 125. 50. 180. 60. 245. 70. Braking Distance (feet) Source: British Highway Code. MULTIPLY MONOMIALS An expression like 210 s2 is called a monomial. A monomial is a number, a variable, or a product of a number and one or more variables. An expression involving the division of variables is not a monomial. Monomials that are real numbers are called constants .. Example 1 Identify Monomials Determine whether each expression is a monomial. Explain your reasoning.. Study Tip Reading Math. Expression. Monomial?. Reason. a.. 5. yes. 5 is a real number and an example of a constant.. b.. pq. no. The expression involves the addition, not the product, of two variables.. c.. x. yes. Single variables are monomials.. d.. c. no. The expression is the quotient, not the product, of two variables.. e.. ab c8 5. yes. abc 8 1 abc 8. The expression is the product of a 5 5 1 number, , and three variables. 5. d. Recall that an expression of the form xn is called a power and represents the product you obtain when x is used as a factor n times. The number x is the base, and the number n is the exponent.. The expression x n is read x to the nth power.. exponent. 25 ←. base. 5 factors. . does doubling speed quadruple braking distance? Ask students: • What does the term quadrupled mean? increased by 4 times, or multiplied by 4 • Why isn’t the braking distance 4 times the speed? The expression for finding the braking distance, 1 where s is speed in mph, is s2, 20 not 4s. • Based on the chart, what would be the braking distance for a car traveling 80 mph? 320 ft 120 mph? 720 ft • Drag Racing Suppose a sports car on a drag strip can reach 100 miles per hour in a quarter of a mile. Using the braking distance table, calculate how far the car would travel on the drag strip, from start to stop, if the driver started braking when the car reached 100 miles per hour. A mile is 5280 feet. 1820 ft. does doubling speed quadruple braking distance? Speed (miles per hour). 5-Minute Check Transparency 8-1 Use as a quiz or review of Chapter 7.. • Multiply monomials.. ←. 1 Focus. Multiplying Monomials. 2 2 2 2 2 or 32. In the following examples, the definition of a power is used to find the products of powers. Look for a pattern in the exponents. 410. Chapter 8 Polynomials. Resource Manager Workbook and Reproducible Masters Chapter 8 Resource Masters • Study Guide and Intervention, pp. 455–456 • Skills Practice, p. 457 • Practice, p. 458 • Reading to Learn Mathematics, p. 459 • Enrichment, p. 460. Parent and Student Study Guide Workbook, p. 59. Transparencies 5-Minute Check Transparency 8-1 Answer Key Transparencies. Technology AlgePASS: Tutorial Plus, Lesson 18 Interactive Chalkboard.

(12) 3 factors. 4 factors. 2 factors. 5 factors. . 34. 2 Teach. 3 3 3 3 3 3 or 36. . . 32. . 2 2 2 2 2 2 2 2 or. 28. . . . 25. . 23. 2 4 or 6 factors. 3 5 or 8 factors. Building on Prior Knowledge. These and other similar examples suggest the property for multiplying powers.. Product of Powers • Words. To multiply two powers that have the same base, add the exponents.. • Symbols For any number a and all integers m and n, am an am n. • Example a4 a12 a4 12 or a16. Example 2 Product of Powers Simplify each expression.. In Chapter 1, students learned that an algebraic expression is an expression consisting of one or more numbers and variables along with one or more arithmetic operations. All monomials are expressions, but not all expressions are monomials.. a. (5x7)(x6) (5x7)(x6) (5)(1)(x7 x6) Commutative and Associative Properties. Study Tip Power of 1 Recall that a variable with no exponent indicated can be written as a power of 1. For example, x x1 and ab a1b1.. b.. (5 1)(x7 6). Product of Powers. 5x13. Simplify.. MULTIPLY MONOMIALS. In-Class Examples. (4ab6)(7a2b3) (4ab6)(7a2b3) (4)(7)(a a2)(b6 b3) Commutative and Associative Properties 28(a1 2)(b6 3). Product of Powers. 28a3b9. Simplify.. 1 Determine whether each expression is a monomial. Explain your reasoning. a. 17 s This is not a monomial because it involves subtraction, not multiplication.. POWERS OF MONOMIALS You can also look for a pattern to discover the property for finding the power of a power. 3 factors. 5 factors. . Apply rule for Product of Powers.. 410. (z8)3 (z8)(z8)(z8) ←. 42 2 2 2 2 ←. . . (42)5 (42)(42)(42)(42)(42). z8 8 8 z24. Therefore, (42)5 410 and (z8)3 z24. These and other similar examples suggest the property for finding the power of a power.. Power of a Power. Study Tip. • Words. To find the power of a power, multiply the exponents.. • Symbols. For any number a and all integers m and n, (am)n am n.. • Example. (k5)9. . k5 9. or. Simplify ((32)3)2.. To review using a calculator to find a power of a number, see Lesson 1-1.. ((32)3)2 (32 3)2. Simplify.. 36 2. Power of a Power. . 312. or 531,441 Simplify.. www.algebra1.com/extra_examples. 3 4. c. This is a monomial because it is a real number.. Teaching Tip. Power of a Power. (36)2. b. 8f 2g This is a monomial because it is the product of a number and two variables.. d. xy This is a monomial because it is the product of two variables.. k45. Example 3 Power of a Power. Look Back. Power Point®. Lesson 8-1 Multiplying Monomials. 411. Make sure students read the Study Tip in the margin next to Example 2. In order for students to find the products of powers correctly, they must remember that the expression x is understood to mean x1. Suggest that students rewrite variables without exponents with an exponent of 1.. 2 Simplify each expression. a. (r4)(12r7) 12r 11. Teacher to Teacher Patricia Lund. Divide County H.S., Crosby, ND. b. (6cd5)(5c5d2) 30c 6d 7. "The paragraph at the top of page 410 interests students because many of them are preparing to drive. I like to graph this data on a coordinate plane so we review ordered pairs and graphing points. We analyze the graph to determine whether or not it is linear and if it is a function. We also review function notation to name the graph suggested by these points." Lesson 8-1 Multiplying Monomials 411.

(13) POWERS OF MONOMIALS. In-Class Examples. Look for a pattern in the examples below. (xy)4 (xy)(xy)(xy)(xy). Power Point®. 3 Simplify [(23)3]2.. of a cube with a side length s 5xyz. (5xyz)3 125x 3y 3z 3. Teaching Tip. Remind students that when multiple sets of grouping symbols are used in an expression, the outermost set is usually a pair of brackets [ ]. The expression [(xy)2]4 is the same as ((xy)2)4.. (x x x x)(y y y y). (6 6 6)(a a a)(b b b). x 4y4. 63a3b3 or 216a3b3. These and other similar examples suggest the following property for finding the power of a product.. 218 or 262,144. 4 GEOMETRY Find the volume. (6ab)3 (6ab)(6ab)(6ab). Study Tip Powers of Monomials. Power of a Product. Sometimes the rules for the Power of a Power and the Power of a Product are combined into one rule. (ambn)p ampbnp. • Words. To find the power of a product, find the power of each factor and multiply.. • Symbols. For all numbers a and b and any integer m, (ab)m a mb m.. • Example (2xy)3 (2)3x3y3 or 8x3y3. Example 4 Power of a Product GEOMETRY Express the area of the square as a monomial.. 5 Simplify [(8g3h4)2]2(2gh5)4.. Area s2. 65,536g16h 36. . Formula for the area of a square. (4ab)2. 42a2b2. 4ab. s 4ab Power of a Product. 4ab. 16a2b2 Simplify.. Concept Check. The area of the square is 16a2b2 square units.. Simplifying Expressions Janice 4 2 53 simplified (x y ) [2(xy)7 ] into 6. The properties can be used in combination to simplify more complex expressions involving exponents.. 8 x13y 22. Explain whether her 6. simplification is complete.. Simplifying Monomial Expressions. The simplification is not complete because the fraction is not in simplest form.. To simplify an expression involving monomials, write an equivalent expression in which: • each base appears exactly once, • there are no powers of powers, and • all fractions are in simplest form.. Answers 2a. (5m)2 25m 2 2b. The power of a product is the product of the powers. 2c. (3a)2 9a 2 2d. 2(c7)3 2c 21 3. When finding the product of powers with the same base, keep the same base and add the exponents. Do not multiply the bases. 4. 5 7d shows subtraction, not multiplication. 4a 3b. 5. shows division, not multiplication. 6. A single variable is a monomial.. Example 5 Simplify Expressions 2. Simplify xy4 [(6y)2]3. 1 3. 2. 13xy4. 2 1 3 1 2 x2(y4)2(6)6y6 3 1 2 8 x y (46,656)y6 9 1 (46,656)x2 y8 y6 9. [(6y)2]3 xy4 (6y)6. . 412. 5184x2y14. Power of a Power Power of a Product Power of a Power Commutative Property Product of Powers. Chapter 8 Polynomials. Differentiated Instruction Logical Give students a term such as 144a10b8 and challenge them to write 20 unique combinations of monomials that would produce this product if multiplied.. 412. Chapter 8 Polynomials.

(14) Concept Check. 1a–c. Sample answers are given. 1a. n 2(n 5) n 7 1b. (n 2)5 n10 1c. (nm 2)5 n 5n10 2a–d. See margin for explanations.. 1. OPEN ENDED Give an example of an expression that can be simplified using each property. Then simplify each expression. a. Product of Powers. b. Power of a Power. c. Power of a Product. 2. Determine whether each pair of monomials is equivalent. Explain. a. 5m2 and (5m)2 no. b. (yz)4 and y4z4 yes. c. 3a2 and (3a)2 no. d. 2(c7)3 and 8c21 no. Study Notebook. 3. FIND THE ERROR Nathan and Poloma are simplifying (52)(59).. Nathan. Poloma. (52)(59) = (5 . 5)2 + 9. (5 2 )(5 9 ) = 5 2. = 2511. + 9. = 511. Who is correct? Explain your reasoning. Poloma; see margin for explanation.. Guided Practice Exercises. Examples. 4–6 7–12 13, 14. 1 2, 3, 5 4. 4a 5. no. 6. n yes. 3b. 4–6. See margin for explanations.. Simplify.. 7. x(x4)(x6) x 11 10.. Application. 8. (4a4b)(9a2b3) 36a6b 4. (3y5z)2 9y10z2. 9. [(23)2]3 218 or 262,144. 11. (4mn2)(12m2n). 12. (2v3w4)3(3vw3)2. 48m 3n 3. 72v11w18. GEOMETRY Express the area of each triangle as a monomial.. 5n 5. 13.. 6a 5b 6. 14. 4ab 5. 2n 2 5n 3. 3a 4b. ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 15–20 21–48 49–54. 1 2, 3, 5 4. Extra Practice See page 837.. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 8. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Determine whether each expression is a monomial. Write yes or no. Explain. 4. 5 7d no. GUIDED PRACTICE KEY. 3 Practice/Apply. FIND THE ERROR Both Nathan and Poloma added the exponents 2 and 9, which is the correct procedure when finding a product of powers. However, tell students to notice how Nathan and Poloma handled the bases. Suggest that students substitute a variable for 5 and then examine which method is correct.. Determine whether each expression is a monomial. Write yes or no. Explain. 15. 12 yes. 16. 4x3 yes. 17. a 2b no. 18. 4n 5m no. x 19. 2 y. 1 20. abc14 yes. no 15–20. See margin for explanations.. About the Exercises…. 5. Organization by Objective • Multiply Monomials: 15–26 • Powers of Monomials: 27–54. Simplify.. 21. (ab4)(ab2) a 2b 6. 22. (p5q4)(p2q) p7q 5. 23. (7c3d4)(4cd3) 28c 4d 7 25.. (5a2b3c4)(6a3b4c2). 27.. (9pq7)2. 30a 5b7c 6. 81p 2q14. 24. (3j7k5)(8jk8) 24j 8k 13 26. (10xy5z3)(3x4y6z3) 30x 5y 11z 6. 29. [(32)4]2 316 or 43,046,721. 0.25x 6. 30. [(42)3]2 412 or 16,777,216. 27 64c 3 35. (4cd)2(3d2)3 432c 2d 8. 32. (0.4h5)3 0.064h 15 4 2 16 34. a2 a 4 5 25 36. (2x5)3(5xy6)2 200x 17y 12. 37. (2ag2)4(3a2g3)2 144a 8g14. 38. (2m2n3)3(3m3n)4 648m18n 13. 31.. (0.5x3)2. 3 3 33. c 4. ★ 39.. . (8y3)(3x2y2). 3 xy4 8. Odd/Even Assignments Exercises 15–42 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 28. (7b3c6)3 343b 9c 18. . 9x 3y 9. ★ 40.. www.algebra1.com/self_check_quiz. 47m (49m)(17p)314p5 2. Lesson 8-1 Multiplying Monomials. Answers 15. 12 is a real number and therefore a monomial. 16. 4x3 is the product of a number and three variables. 17. a 2b shows subtraction, not multiplication of variables.. Assignment Guide. 8m 3p 6. 18. 4n 5m shows addition, not multiplication of variables.. 413. Basic: 15–37 odd, 43–47 odd, 49–52, 55–82 Average: 15–47 odd, 51–82 Advanced: 16–50 even, 53–74 (optional: 75–82). x y. 19. 2 shows division, not multiplication of variables. 1 5. 20. abc14 is the product of a 1 5. number, , and several variables. Lesson 8-1 Multiplying Monomials 413.

(15) NAME ______________________________________________ DATE. ★ 41. Simplify the expression (2b3)4 3(2b4)3. 40b 12 ★ 42. Simplify the expression 2(5y3)2 (3y3)3. 50y 6 27y 9. ____________ PERIOD _____. Study Guide andIntervention Intervention, 8-1 6-1 Study Guide and p. 455 (shown) Multiplying Monomialsand p. 456. Multiply Monomials A monomial is a number, a variable, or a product of a number and one or more variables. An expression of the form xn is called a power and represents the product you obtain when x is used as a factor n times. To multiply two powers that have the same base, add the exponents.. Example 2. Simplify (3x6)(5x2).. (3 5)(x6 2) 15x8. (3x6)(5x2). (3)(5)(x6. x2). The product is. Associative Property. Simplify (4a3b)(3a2b5).. (4a3b)(3a2b5). 12(a3 2)(b1 5) 12a5b6. The product is. 12a5b6.. Product of Powers Simplify.. 15x8.. (4)(3)(a3. a2)(b. b5). 43.. Lesson 8-1. Example 1. GEOMETRY Express the area of each figure as a monomial.. For any number a and all integers m and n, am an a m n.. Product of Powers. 15f 5g 5 44.. a 4b 2. 3fg 2. a. 5f 4g 3. 45.. (49x 8). 2b. 7x 4. a 2b. Exercises Simplify. 2. n2 n7. 1. y( y5). y6. 7x 6. 5. m m5. 4. x(x2)(x4). x7. GEOMETRY Express the volume of each solid as a monomial.. 3. (7x2)(x4). n9. 46.. 6. (x3)(x4). m6. 64k 9. x 3y 5 48.. 47.. 2n. x 2y. 4k 3. x7. 16n 5 4n 3. y 7. (2a2)(8a). 8. (rs)(rs3)(s2). 16a 3 1 3. 10. (2a3b)(6b3). 4x 3y 4. 11. (4x3)(5x7). 4a 3b 4. 15. . 4k 3. 6j 3k10. 14. (5xy)(4x2)( y4). 15. (10x3yz2)(2xy5z). 20x 3y 5. 20x 4y 6z 3. a 3b 2c7. 4k 3. 12. (3j 2k4)(2jk6). 20x10. 13. (5a2bc3) abc4. xy3. 9. (x2y)(4xy3). r 2s 6. NAME ______________________________________________ DATE. Skills Practice, 8-1 Practice (Average). TELEPHONES For Exercises 49 and 50, use the following information. The first transatlantic telephone cable has 51 amplifiers along its length. Each amplifier strengthens the signal on the cable 106 times. 49. After it passes through the second amplifier, the signal has been boosted 106 106 times. Simplify this expression. 1012 or 1 trillion. ____________ PERIOD _____. p. 457 and Practice, 458 (shown) Multiplyingp. Monomials. 50. Represent the number of times the signal has been boosted after it has passed through the first four amplifiers as a power of 106. Then simplify the expression. (106)4 or 1024. Determine whether each expression is a monomial. Write yes or no. Explain. 21a2 7b. 1. No; this involves the quotient, not the product, of variables. b3c2 2. 1 2. 2. Yes; this is the product of a number, , and two variables. Simplify. 3. (5x2y)(3x4) 15x 6y. 4. (2ab2c2)(4a3b2c2) 8a 4b 4c 4. 5. (3cd4)(2c2) 6c 3d 4. 6. (4g3h)(2g5) 8g 8h. . . 1 3. 7. (15xy4) xy3 5x2y 7. 8. (xy)3(xz) x 4y 3z. . 10. (0.2a2b3)2 0.04a 4b 6. . 1 6. 9. (18m2n)2 mn2 54m 5n 4. 23 . 11. p. 2. 14 . 4 p 2 9. 12. cd3. 13. (0.4k3)3 0.064k 9. 2. DEMOLITION DERBY For Exercises 51 and 52, use the following information. When a car hits an object, the damage is measured by the collision impact. For a certain car, the collision impact I is given by I 2s2, where s represents the speed in kilometers per minute.. 1 c 2d 6 16. 14. [(42)2]2 48 or 65,536. 51. What is the collision impact if the speed of the car is 1 kilometer per minute? 2 kilometers per minute? 4 kilometers per minute? 2; 8; 32. GEOMETRY Express the area of each figure as a monomial. 15.. 16.. 17. 5x 3. 3ab 2. 6ac 3. 6a 2b 4. 52. As the speed doubles, explain what happens to the collision impact.. 4 a 2c. 18a 3b 6. (25x 6). The collision impact quadruples, since 2(2s)2 is 4(2s 2).. 12a 3c 4. GEOMETRY Express the volume of each solid as a monomial. 18.. 19.. 20.. n. 3h 2. 3g. mn 3. TEST TAKING For Exercises 53 and 54, use the following information. A history test covers two chapters. There are 212 ways to answer the 12 true-false questions on the first chapter and 210 ways to answer the 10 true-false questions on the second chapter.. 7g 2. m 3n 3h 2 3h 2. 27h 6. m 4n 5. (63g 4). 21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five light switches can set in twice this many ways. In how many ways can five light switches be set? 25 or 32 22. HOBBIES Tawa wants to increase her rock collection by a power of three this year and then increase it again by a power of two next year. If she has 2 rocks now, how many rocks will she have after the second year? 26 or 64. NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 8-1 Readingto to Learn Learn Mathematics. ELL. Mathematics, p. 459 Multiplying Monomials. Pre-Activity. Why does doubling speed quadruple braking distance? Read the introduction to Lesson 8-1 at the top of page 410 in your textbook. Find two examples in the table to verify the statement that when speed is doubled, the braking distance is quadrupled. Write your examples in the table. Speed (miles per hour). Braking Distance (feet). Speed Doubled (miles per hour). Braking Distance Quadrupled (feet). 20. 20. 40. 80. 30. 45. 60. 180. Demolition Derby In a demolition derby, the winner is not the car that finishes first but the last car still moving under its own power.. The monomial 3xy is the product of the constant 3 and the variables x and y.. 54. If a student guesses on each question, what is the probability of answering all questions correctly? 1 4,194,304. Source: Smithsonian Magazine. CRITICAL THINKING Determine whether each statement is true or false. If true, explain your reasoning. If false, give a counterexample.. 55–57. See margin for explanations or counterexamples.. 55. For any real number a, (a)2 a2. false. Reading the Lesson 1. Describe the expression 3xy using the terms monomial, constant, variable, and product.. 53. How many ways are there to answer all 22 questions on the test? (Hint: Find the product of 212 and 210.) 222 or 4,194,304 ways. 414. 56. For all real numbers a and b, and all integers m, n, and p, (ambn)p ampbnp. true 57. For all real numbers a, b, and all integers n, (a b)n an bn. false. Chapter 8 Polynomials. 2. Complete the chart by choosing the property that can be used to simplify each expression. Then simplify the expression. Expression. Property. Expression Simplified. Product of Powers 35 32. Power of a Power. 37 or 2187. NAME ______________________________________________ DATE. 8-1 Enrichment Enrichment,. ____________ PERIOD _____. p. 460. Power of a Product Product of Powers (a 3)4. Power of a Power. a12. Power of a Product Product of Powers (4xy)5. Power of a Power. 1024x 5y 5. Power of a Product. Helping You Remember 3. Write an example of each of the three properties of powers discussed in this lesson. Then, using the examples, explain how the property is used to simplify them.. Sample answer: For z 2 z 5, since the bases are the same, use the Product of Powers Property and add the exponents to get z 7. For (a 4)3, use the Power of a Power Property. Multiply the exponents to get a12. For (3rs)3, use the Power of a Product Property. Raise the constant and each variable to the power to get 27r 3s 3.. An Wang An Wang (1920–1990) was an Asian-American who became one of the pioneers of the computer industry in the United States. He grew up in Shanghai, China, but came to the United States to further his studies in science. In 1948, he invented a magnetic pulse controlling device that vastly increased the storage capacity of computers. He later founded his own company, Wang Laboratories, and became a leader in the development of desktop calculators and word processing systems. In 1988, Wang was elected to the National Inventors Hall of Fame. Digital computers store information as numbers. Because the electronic circuits of a computer can exist in only one of two states, open or closed, the numbers that are stored can consist of only two digits, 0 or 1. Numbers written using only these two digits are called binary numbers. To find the decimal value of a binary number, you use the digits to write a polynomial in 2. For instance, this is how to find the decimal value of the number 10011012. (The subscript 2 indicates that this is a binary number.) 10011012 1 26 0 25 0 24 1 23 1 22 0 21 1 20 1 6 40 32 16 8 1 4 02 1 0 1 1. 414. Chapter 8 Polynomials.

(16) 58. WRITING IN MATH. Answer the question that was posed at the beginning of the lesson. See margin.. 4 Assess. Why does doubling speed quadruple braking distance? Include the following in your answer: • the ratio of the braking distance required for a speed of 40 miles per hour and the braking distance required for a speed of 80 miles per hour, and 1 1 • a comparison of the expressions s2 and (2s)2. 20. Standardized Test Practice. 20. 59. 42 45 ? D A. 167. B. 87. C. 410. D. 47. 60. Which of the following expressions represents the volume of the cube? D A C. 15x3 25x3. B D. 25x2 125x3 5x. Mixed Review. Solve each system of inequalities by graphing. (Lesson 7-5) 61. y 2x 2 y x 1. 62. y y. x2 2x 1. 2 x3. 63. x y. Use elimination to solve each system of equations. (Lesson 7-4) 64. 4x 5y 2 x 2y 6 (2, 2). 65. 3x 4y 25 66. x y 20 (4, 16) 2x 3y 6 (3, 4) 0.4x 0.15y 4. Solve each compound inequality. Then graph the solution set. (Lesson 6-4). 67–70. See margin for graphs.. 67. 4 h 3 or 4 h 69. 14. Modeling Write an expression such as (2x3y)3(3x2y4) 2 on the chalkboard, but write the exponents on self-adhesive notes. Ask student volunteers to simplify the expression, using more selfadhesive notes to show the multiplication and addition of exponents in each step of the simplification process.. Getting Ready for Lesson 8-2. Maintain Your Skills 61–63. See pp. 471A–471B.. Open-Ended Assessment. 3h 2. 2. 5 {h |h. or h. 7 68. 4 4a 12 24 {a|2 a 3} 1} 70. 2m 3 7 or 2m 7 9 {m|m 1}. Answers. Determine whether each transformation is a reflection, translation, dilation, or rotation. (Lesson 4-2) 71.. 72.. reflection. 73.. PREREQUISITE SKILL Students will learn about dividing monomials in Lesson 8-2. Part of dividing monomials involves simplifying fractions formed by dividing coefficients. Use Exercises 75–82 to determine your students’ familiarity with simplifying fractions.. 58. Answers should include the following. 80 feet 320 feet. • the ratio , which. dilation. simplifies to a ratio of 1 to 4 • If s is replaced by 2s in the formula for the breaking distance required for a car traveling s miles per hour the result is. rotation. 74. TRANSPORTATION Two trains leave York at the same time, one traveling north, the other south. The northbound train travels at 40 miles per hour and the southbound at 30 miles per hour. In how many hours will the trains be 245 miles apart? (Lesson 3-7) 1 3 h 2. Getting Ready for the Next Lesson. PREREQUISITE SKILL Simplify. 3 1 2 1 75. 76. 15 5 6 3 14 7 9 3 79. 80. 36 18 48 16. 1 (2s)2. Using the Power of a 20. Product and Power of a Power Properties, this simplifies to. 201 . (To review simplifying fractions, see pages 798 and 799.). 10 77. 2. 5 44 11 81. 32 8. 4 s 2 . This means that. 27 78. 3. 9 45 5 82. 18 2. Lesson 8-1 Multiplying Monomials. doubling the speed of car multiplies the breaking distance by 4.. 415. 67.. Answers 55. Let a 2 and b 3. Then (ab)2 (2 3)2 or 36 and ab 2 (2)(3)2 or 18. 56. (amb n )p (a m )p(b m)p Power of a Product ampbmp Power of a Power 57. Let a 3, b 4, and n 2. Then (a b)n (3 4)2 or 49 and an bn 32 42 or 25.. 8. 6. 4. 2. 0. 2. 68. 4. 2. 0. 2. 4. 6. 4. 2. 0. 2. 4. 6. 4. 2. 0. 2. 4. 6. 69. 70.. Lesson 8-1 Multiplying Monomials 415.

(17) Algebra Activity. A Follow-Up of Lesson 8-1. A Follow-Up of Lesson 8-1. Investigating Surface Area and Volume. Getting Started. Collect the Data. 3 cm • Cut out the pattern shown from a sheet of centimeter grid 2 cm paper. Fold along the dashed lines and tape the edges together to form a rectangular prism with dimensions 2 centimeters by 5 centimeters by 3 centimeters. 5 cm • Find the surface area SA of the prism by counting the squares on all the faces of the prism or by using the formula SA 2w 2wh 2h, where w is the width, is the length, and h is the height of the prism. 62 cm2 • Find the volume V of the prism by using the formula V wh. 30 cm3 • Now construct another prism with dimensions that are 2 times each of the dimensions of the first prism, or 4 centimeters by 10 centimeters by 6 centimeters. • Finally, construct a third prism with dimensions that are 3 times each of the dimensions of the first prism, or 6 centimeters by 15 centimeters by 9 centimeters.. Objective Determine surface area ratio and volume ratio of rectangular prisms when dimensions are multiplied by a. Materials centimeter grid paper scissors tape. Teach • Any size grid paper will work for this activity because the units are not important. • Before students begin this activity, direct their attention to the data table. Ask students to make a conjecture about the surface area and volume of the prism when each dimension is multiplied by two, and by three. Write student predictions on the chalkboard to use in a discussion after the activity has been completed. • Have students verify the formula for the surface area of a prism by counting the squares on the surface of their prism.. Analyze the Data 1. Copy and complete the table using the prisms you made.. Study Notebook You may wish to have students summarize this activity and what they learned from it.. 416. Chapter 8 Polynomials. Dimensions. Surface Area (cm2). Volume (cm3). Original. 2 by 5 by 3. 62. 30. A. 4 by 10 by 6. 248. 240. B. 6 by 15 by 9. 558. 810. Surface Area Ratio. Volume Ratio. SA of New SA of Original . V of New V of Original . 248 4 62 558 9 62. 240 8 30 810 27 30. 2. Make a prism with different dimensions from any in this activity. Repeat the steps. in Collect the Data, and make a table similar to the one in Exercise 1. See pp. 471A–471B.. Make a Conjecture 3. Suppose you multiply each dimension of a prism by 2. What is the ratio of the. surface area of the new prism to the surface area of the original prism? What is the ratio of the volumes? 4; 8 4. If you multiply each dimension of a prism by 3, what is the ratio of the surface area of the new prism to the surface area of the original? What is the ratio of the volumes? 9; 27 5. Suppose you multiply each dimension of a prism by a. Make a conjecture about the ratios of surface areas and volumes. a 2; a 3. Assess Students should determine that if the length, width, and height of a rectangular prism are each multiplied by a, the resulting surface area will be a2 times the original surface area, and the resulting volume will be a3 times the original volume.. Prism. Extend the Activity 6. Repeat the steps in Collect the Data and Analyze the Data using cylinders. To. start, make a cylinder with radius 4 centimeters and height 5 centimeters. To compute surface area SA and volume V, use the formulas SA 2 r2 2 rh and V r2h, where r is the radius and h is the height of the cylinder. Do the conjectures you made in Exercise 5 hold true for cylinders? Explain. See pp. 471A–471B.. 416. Chapter 8 Polynomials. Resource Manager Teaching Algebra with Manipulatives. Glencoe Mathematics Classroom Manipulative Kit. • p. 2 (master for centimeter grid paper) • p. 135 (student recording sheet). • scissors.

(18) Lesson Notes. Dividing Monomials • Simplify expressions involving the quotient of monomials.. 1 Focus. • Simplify expressions containing negative exponents. 0. Vocabulary. can you compare pH levels?. Battery acid. 1. Lemon juice. 2. • zero exponent • negative exponent. To test whether a solution is a base or an acid, chemists use a pH test. This test measures the concentration c of hydrogen ions (in moles per liter) in the solution.. Increasing acidity. Tomatoes. 4. Coffee. 5 6. Neutral. 1 pH c 10. . Pure water Baking soda. 9. Increasing alkalinity. Mathematical Background notes are available for this lesson on p. 408C.. Milk. 7 8. The table gives examples of solutions with various pH levels. You can find the quotient of powers and use negative exponents to compare measures on the pH scale.. 5-Minute Check Transparency 8-2 Use as a quiz or review of Lesson 8-1.. Vinegar. 3. Milk of Magnesia. 10 11. can you compare pH levels? Ask students: 1 pH • Using the formula c ,. Ammonia. 12. Bleach. 13. Lye. 14. 10 . Source: U.S. Geological Survey. QUOTIENTS OF MONOMIALS In the following examples, the definition of a power is used to find quotients of powers. Look for a pattern in the exponents. 5 factors. 6 factors. . . 1. 1. 1. 1. 45 44444 4 4 or 42 43 444. 36 3 3 3 3 3 3 3 3 3 3 or 34 32 33 1. . . 1. 3 factors. 1. 5 3 or 2 factors. 1. . 1. . 1. 2 factors. 6 2 or 4 factors. These and other similar examples suggest the following property for dividing powers.. Quotient of Powers • Words. To divide two powers that have the same base, subtract the exponents. m. m n. • Symbols For all integers m and n and any nonzero number a, aa n a 15. b b15 7 or b8 • Example b7. TEACHING TIP Ask students why a and b cannot be 0.. Example 1 Quotient of Powers. what is the concentration of hydrogen ions in a solution with pH 1? 0.1 moles/liter • Using the same formula, what is the concentration of hydrogen ions in a solution with pH 2? 0.01 moles/liter • As the pH increases by 1, what happens to the hydrogen ion concentration? It gets 10 times as small. • According to the scale, ammonia has a pH of 12. Would this indicate that ammonia has a very large or very small concentration of hydrogen ions? a very small concentration. a5b8 ab. Simplify 3 . Assume that a and b are not equal to zero. a5b8 a5 b8 3 ab3 a b. . Group powers that have the same base.. (a 5 1)(b8 3). Quotient of Powers. . Simplify.. a 4b 5. Lesson 8-2 Dividing Monomials. 417. Resource Manager Workbook and Reproducible Masters Chapter 8 Resource Masters • Study Guide and Intervention, pp. 461–462 • Skills Practice, p. 463 • Practice, p. 464 • Reading to Learn Mathematics, p. 465 • Enrichment, p. 466 • Assessment, p. 517. Parent and Student Study Guide Workbook, p. 60. Transparencies 5-Minute Check Transparency 8-2 Answer Key Transparencies. Technology AlgePASS: Tutorial Plus, Lesson 19 Interactive Chalkboard. Lesson x-x Lesson Title 417.

(19) In the following example, the definition of a power is used to compute the power of a quotient. Look for a pattern in the exponents.. 2 Teach. 3 factors. . 2 3 2 2 2 222 23 or 5 5 5 5 555 53. Remind students that multiplying and dividing are inverse or opposite operations. With that in mind, ask students to make a conjecture about dividing monomials based on what they know about multiplying monomials.. . . . . Building on Prior Knowledge. 3 factors. 3 factors. This and other similar examples suggest the following property.. Power of a Quotient • Words. To find the power of a quotient, find the power of the numerator and the power of the denominator.. • Symbols. For any integer m and any real numbers a and b, b. • Example. dc. 5. m. m. a a . 0, m b. c5 d. b. 5. QUOTIENTS OF MONOMIALS. In-Class Examples. Example 2 Power of a Quotient. Power Point®. 2p2 4. 3. Simplify .. x7y12. 1 Simplify 6 3 . Assume that. 2p2 4. xy. 3. x and y are not equal to zero. xy 9. (2p2)4 3. 4. Power of a Quotient. 24(p2)4 3. Power of a Product 4. Teaching Tip. Remind students not to forget to find the powers of the constant terms of the monomials.. 16p8 81. . 4c d 2 Simplify 4 7 . Assume that. Power of a Power. 3 2 3. NEGATIVE EXPONENTS A graphing calculator can be used to investigate. 5e f. expressions with 0 as an exponent as well as expressions with negative exponents.. e and f are not equal to zero. 64c 9d 6 125e12f 21. Zero Exponent and Negative Exponents Use the. Study Tip Graphing Calculator. key on a TI-83 Plus to evaluate expressions with exponents. 1 Think and Discuss 2a–d. They are reciprocals. 3. 5 1. Copy and complete the table below.. To express a value as a fraction, press. Power. ENTER. 2.. ENTER .. 3. 4. 5. 418. 24. 23. 22. 21. 20. 16. 8. 4. 2. 1. 21. 22. 23. 24. 1 1 1 1 2 4 8 16 Describe the relationship between each pair of values. b. 23 and 23 c. 22 and 22 d. 21 and 21 a. 24 and 24 1 Make a Conjecture as to the fractional value of 5 . Verify your conjecture using a calculator. What is the value of 50? 1 What happens when you evaluate 00? An error message appears. Value. Chapter 8 Polynomials. Zero and Negative Exponents Make sure students understand that 24 and 24 are reciprocals of each other. After answering Exercise 4, ask students to write an expression involving division 06 0. 06 0. 0 0. that is equivalent to 00. A sample answer is 6 . Show students that 6 . and remind them that division by zero is undefined. 418. Chapter 8 Polynomials.

(20) To understand why a calculator gives a value of 1 for 20, study the two methods 24 2. used to simplify 4 .. Study Tip. Method 1. Alternative Method Another way to look at the. 24 24 4 24. problem of simplifying 4 2 is to recall that any nonzero number divided. 20. 24. 24 2. 16 16. by itself is 1: 4 or 1.. In-Class Example. Method 2 1. 1. 1. 1. 1. 1. Subtract.. 1. Ask students why the negative sign does not affect the outcome. Students should explain that any nonzero number raised to the zero power is 1, and a negative number is a nonzero number.. Definition of powers. 1. Simplify.. 24. Since 4 cannot have two different values, we can conclude that 20 1. 2. Zero Exponent • Words. Power Point®. Teaching Tip. 1. 24 2222 24 2222. Quotient of Powers. NEGATIVE EXPONENTS. Any nonzero number raised to the zero power is 1.. 3 Simplify each expression. Assume that m and n are not equal to zero.. • Symbols For any nonzero number a, a0 1. • Example (0.25)0 1. n 12m 8m n . 8 7 0. a. 5 10. 1. m0n3 n. Example 3 Zero Exponent. b. n 2. Simplify each expression. Assume that x and y are not equal to zero. 3x5y 0 8xy. . a. 7. . 3x5y 0. 8x y 7. 1 a0 1. 3 0. ts b. . t t3s0 t3(1) t t t3 t. . a0 1 Simplify.. t2. Quotient of Powers. To investigate the meaning of a negative exponent, we can simplify expressions 82 8. like 5 in two ways. Method 1 82. 82 5 85. 83. Method 2 1. 1. 88 85 88888 82. Quotient of Powers. 1. 1 8. 3. Subtract.. Definition of powers. 1. Simplify.. 82. 1 8. Since 5 cannot have two different values, we can conclude that 83 3 . 8. TEACHING TIP Ask students to explain why 4a2 is not a monomial. They should make the connection that an expression involving a negative exponent in the numerator is not a monomial.. Negative Exponent • Words. For any nonzero number a and any integer n, an is the reciprocal of an. In addition, the reciprocal of an is an.. • Symbols. 1 1 an. For any nonzero number a and any integer n, an n and n. • Examples. 1 1 52 2 or . a. 5. 25. a. Chalkboard. 1 3 m3 m. www.algebra1.com/extra_examples. PowerPoint® Presentations Lesson 8-2 Dividing Monomials. 419. Differentiated Instruction Naturalist When one-celled organisms reproduce, the population increases by a factor of 2. Population can be counted by multiplying by powers of 2. Use this pattern to show values of negative powers of 2.. 2 23 8. 2 22 4. 2. Interactive. 2 21 2. 2. 20 1 2. 21 ?. 22 ?. 23 ?. 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 8-2 Dividing Monomials 419.

(21) In-Class Examples. An expression involving exponents is not considered simplified if the expression contains negative exponents.. Power Point®. Teaching Tip. Point out to students that rewriting a polynomial as a product of fractions makes applying the Negative Exponent Property easier. Fractions that have negative exponents can be rewritten as their reciprocals.. Example 4 Negative Exponents Simplify each expression. Assume that no denominator is equal to zero. 3 2. b c a. 5. d b3c2 b3 c2 1 5 d 5 1 1 d 1 c2 d5 3 b 1 1 c2d5 b3. . x6. y4. a. 6 9 4 9. y z x z 75p3q5 5r 8 b. 5 4 8 p 2q 15p q r. Teaching Tip. Make sure students realize that the diameter of the circle is the same measure as the length of a side of the square. It is very important that students recognize that the area of the square is (2r)2 and not 2r2.. . C 2r. 1 3. 4. 1 1 7 a. Negative Exponent and Zero Exponent Properties. c5 7a. Multiply fractions.. Example 5 Apply Properties of Exponents Multiple-Choice Test Item Write the ratio of the area of the circle to the area of the square in simplest form. A. 2 r 2 r D 1. 3. A factor is moved from the numerator of a fraction to the denominator or vice versa only if the exponent of the factor is 1 negative; 4 .. Quotient of Powers and Negative Exponent Properties Simplify.. 6. Standardized Test Practice. Group powers with the same base.. 1 7. 6 (1)c5. B . Answer. . a6b0c5. figure in Example 5 of the Student Edition. Write the ratio of the circumference of the circle to the area of the square in simplest form. C 2r. Multiply fractions.. 4 7. Do not confuse a negative number with a number raised to a negative power. 3. 1 a. an n. 21a b c 3a4b7 3 a4 b7 1 21a2b7c5 21 a2 b7 c5 1 (a4 2)(b7 7)(c5) 7. Common Misconception. 1 31 3. Write as a product of fractions.. 3a b b. 2 7 5. Study Tip. 5 TEST ITEM Refer to the. A. . . 4 Simplify each expression. Assume that no denominator is equal to zero.. . 2. 4. B. C. 2 1. D. Read the Test Item A ratio is a comparison of two quantities. It can be written in fraction form.. Test-Taking Tip. Solve the Test Item. Some problems can be solved using estimation. The area of the circle is less than the area of the square. Therefore, the ratio of the two areas must be less than 1. Use 3 as an approximate value for to determine which of the choices is less than 1.. • area of circle r2 length of square diameter of circle or 2r area of square (2r)2 2. area of circle r • 2 area of square. Substitute.. (2r). r 2 2. Quotient of Powers. 4. r 0 or r 0 1 4. 4. The answer is B. 420. Chapter 8 Polynomials. Example 5 Remind students that most of the answer choices in multiple-choice test items are answers that result from an arithmetic or other mistake when solving the problem. For example, answer choice A in Example 5 is , which is the 2 answer you get if you use 2r2 as the area of the square.. Standardized Test Practice. 420. Chapter 8 Polynomials. 3. r.

(22) Concept Check 1. Sample answer: 9xy and 6xy2 a 3b 5 ab. 3 1 5 2 2. 2 a a b b. . 3 Practice/Apply. 1. OPEN ENDED Name two monomials whose product is 54x2y3. a3b5. 2. Show a method of simplifying using negative exponents instead of the ab2 Quotient of Powers Property. 4x3 x. Study Notebook. 3. FIND THE ERROR Jamal and Emily are simplifying 5 .. a 31b 52 a 2b 3. Jamal. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 8. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Emily. –4x3 = –4x3 – 5 x5. –4x3 x3–5 = 4 x5 x–2 4 1 = 2 4x. = –4x–2. = . –4 x. = 2. Who is correct? Explain your reasoning. Jamal; see margin for explanation.. Guided Practice. Simplify. Assume that no denominator is equal to zero. x8y12 xy. 8. GUIDED PRACTICE KEY Exercises. Examples. 4–12 13. 1–4 5. Standardized Test Practice. 7 4. 2 76 or 117,649. 6 5 5. 2 7 x y. 7. 1 7. y0(y5)(y9) 4 y q4 5pq7 10. 3 6 10p q 2p 5. 1 8. 132 169 2 3. (cd ) 11 12 11. 4 9 2 c d (c d ). 2c3d 3 8c 9d 3 6. 7z2 343z 6 5 g8 c 9. d3g8 d 3c 5 (4m3n5)0 1 12. mn mn. 13. Find the ratio of the volume of the cylinder to the volume of the sphere. C 1 2 3 2. A C. B. Volume of sphere 43 πr 3. x. 2x. 1. 1 4. accidentally rewrote 4 as .. Volume of cylinder πr 2h. 3 2. D. FIND THE ERROR It appears that Emily was so busy thinking about what to do with negative exponents that she Remind students that a negative number is different from a number raised to a negative exponent.. ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 14–21 22–37. 1, 2 1–4. Extra Practice See page 837.. Simplify. Assume that no denominator is equal to zero. 12. 4 10 or 1,048,576 14. 2 4 4. y3z9 17. yz2. y 2z 7. 13. 3 6 15. 7 3 or 729 3. 18.. 5b4n 2 25b8n2 2a6 4a12. . . p7n3 pn. 16. 42 p 3n 19.. . 1 15b 21. 5 4 45b 3b. 1 22. x3y0x7 4 x. 1 23. n2(p4)(n5) 34 n p. 1 24. 62 36. 1 25. 53 125. 2. 5. 3. 25 16. 3 27. 2. 7 4. 8 27. 28a c 4 4 28. 3 0 8 4a c 7a b c. 30h2k14 6k17 29. 5hk3 h3. 30. 9xy 3z 6 2. 19y0z4 19 31. 3z16 3z12. 1 (5r2)2 32. (2r3)2 100r 2. p4q3 p 33. (p5q2)1 q. r t 34. 1 1. 2. 0. 4c d 35. 2 3 1 1 b cd. www.algebra1.com/self_check_quiz. 18x3y4z7 2x yz. ★ 36.. 2 4 1. 5b n n2z3 . Organization by Objective • Quotients of Monomials: 14–21, 38–39 • Negative Exponents: 22–37, 40–45. . 1 2a3 20. 5 10a8 5a. 4 26. . About the Exercises…. 3m7 4 81m 28 53 4x y 256x 20y12. 2 5 0. t. b2 2 5n z 3. ★ 37.. 2. 1 3. 2a bc 3ab2 . 27a 9c 3 8b 9. Lesson 8-2 Dividing Monomials. 421. Unlocking Misconceptions • Powers of Negative Numbers Students may assume that the expression 63 means (6)(6)(6). Explain that 63 means (63). To express 6 to the third power, they must use parentheses, (6)3. • Zero Exponents The Zero Exponent property states that any nonzero number raised to the zero power is equal to 1. So, (6)0 1. However, (60) 1. The expression (60) means the opposite of 6 raised to the zero power. So, the opposite of 1 is 1.. Odd/Even Assignments Exercises 14–39 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! You may wish to have students use calculators to aid them in computation of numerical powers.. Assignment Guide Basic: 15–35 odd, 39–42, 47–77 Average: 15–39 odd, 40–44, 47–77 Advanced: 14–38 even, 43–71 (optional : 72–77) Lesson 8-2 Dividing Monomials 421.

(23) NAME ______________________________________________ DATE. ____________ PERIOD _____. 38. The area of the rectangle is 24x5y3 square units. Find the length of the rectangle. 3x2y units. Study Guide andIntervention Intervention, 8-2 6-2 Study Guide and p. 461 (shown) Dividing Monomials and p. 462. Quotients of Monomials. To divide two powers with the same base, subtract the. 39. The area of the triangle is 100a3b square units. Find the height of the triangle. 10ab units. exponents. am a. Quotient of Powers. For all integers m and n and any nonzero number a, am n. n. Power of a Quotient. For any integer m and any real numbers a and b, b. a4b7 ab. Example 2. Simplify . Assume 2. neither a nor b is equal to zero.. . a4b7 a4 b7 2 a b ab2. am b. . m. 8x 3y 2. . 2a3b5 3 3b. . Simplify . 2. Assume that b is not equal to zero.. . Group powers with the same base.. (a4 1)(b7 2) a3b5 The quotient is a3b5 .. m. . 2a3b5 3 (2a3b5)3 3b2 (3b2)3 23(a3)3(b5)3 (3)3(b2)3. Quotient of Powers Simplify.. 20a 2. Power of a Quotient. Power of a Product. 8a9b15 27b. Power of a Power. 8a9b9 27. Quotient of Powers. 6 8a9b9 27. The quotient is .. Lesson 8-2. Example 1. ab . 0, . SOUND For Exercises 40–42, use the following information. The intensity of sound can be measured in watts per square meter. The table gives the watts per square meter for some common sounds.. Exercises Simplify. Assume that no denominator is equal to zero. 55 5. 2. 4 m2. m6 m. a2. x5 y3. 1. 2 53 or 125. 4. a. 2y7. x y. xy6 y x. 8a 3b 3. 3r2r ss . 81 r 4s 8 16. 8. . 2vv ww . 3 4. 10. 4 3. 14y. 2aa b 2. 7. y2 4. 3. 6 3 4. 16v 4. 1 7. 6. y 2 5. 5. y 5 2. a. 5. p5n4 p n. 3. p 3n 3 2. 11. 5. q 4p 3p q . 4 4 3. 9. 2 2. 64 p 6q 6 27. r7s7t 2 s r t. 12. r 4s 4 3 3 2. NAME ______________________________________________ DATE. Skills Practice, 8-2 Practice (Average). ____________ PERIOD _____. p. 463 and Practice, p. 464 (shown) Dividing Monomials. Watts/Square Meter. Common Sounds. 102 101 100 10-2 10-3 10-6 10-7 10-9 10-12. jet plane (30 m away) pain level amplified music (2 m away) noisy kitchen heavy traffic normal conversation average home soft whisper barely audible. Simplify. Assume that no denominator is equal to zero. 88 8. 2. a 3b 3 3. a4b6 ab. m5np m p. 5. 2. 1. 4 84 or 4096. 4f3hg 3. 7p6ws . 64f 9g 3 27h18. 3. 7. 6. 5. 8. 6 3. 37 . 2. 6. 2yz 6 5. 43 . 49 9. 4. 14. . 15w0u1 5u. 2. 36w10 49p12s 6. 9. 3 5. 4c2 24c. 12. 122. 1 6c. 1 144. 22r3s2 11r s. 81 256. 5 15. 2 3 2rs. 8c3d2f 4 4c d f. 3 u. 3 5 0. x4 y . 16. 4 3. 4 7 17. 1 2 3 2c f. 18. 3. 6f 2g3h5 g 8h 2 19. 54f 2g5h3 9. 12t1u5v4 20. 2t3uv5. r4 r 21. 3 (3r) 27. m2n5 (m n ). 1 3 5. qqr r . 25. 2. 6t 2u 4 v9. ( j1k3)4 j j k k. m2 n. 22. 4 3 1 2. 3 3 1. 7cc ded . 26. 5 4. 1. (2a2b)3 a 4 5a b 40b. 24. 2 4 7. 23. 3 3 15. q10 r 25. Sound. 8y7z6 4y z. p 11. p(q2)(r3) q 2r 3. 1 10. x3( y5)(x8) x 5y 5. 13. . 5d 2 4. 5c2d3 4c d. 4. mn 4. c8 7d 2e 4. 3x2x yzy z 3 2. 27. 4 2. 2. 9x 2 4y 2z 6. 28. BIOLOGY A lab technician draws a sample of blood. A cubic millimeter of the blood contains 223 white blood cells and 225 red blood cells. What is the ratio of white blood 1 cells to red blood cells? . 484. 29. COUNTING The number of three-letter “words” that can be formed with the English alphabet is 263. The number of five-letter “words” that can be formed is 265. How many times more five-letter “words” can be formed than three-letter “words”? 676. NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 8-2 Readingto to Learn Learn Mathematics Mathematics, p. 465 Dividing Monomials. Pre-Activity. ELL. How can you compare pH levels? Read the introduction to Lesson 8-2 at the top of page 417 in your textbook.. 101 . • In the formula c . 40. How many times more intense is the sound from heavy traffic than the sound from normal conversation? 103 or 1000. xy2 xy. 3. y. pH. , identify the base and the exponent.. 1 base , exponent pH 10. 41. What sound is 10,000 times as loud as a noisy kitchen? jet plane. Timbre is the quality of the sound produced by a musical instrument. Sound quality is what distinguishes the sound of a note played on a flute from the sound of the same note played on a trumpet with the same frequency and intensity.. 42. How does the intensity of a whisper compare to that of normal conversation? 1 1000 PROBABILITY For Exercises 43 and 44, use the following information. 1 If you toss a coin, the probability of getting heads is . If you toss a coin 2 times, 2. 1 2. 44. Express your answer to Exercise 43 as a power of 2. 2n. 1 n 2. 43. . 1 1 10 10 1 1 to cm 100,000 10,000 1 1 46. 7 to 9 cm; 10 10 1 to 10,000,000 1 cm 1,000,000,000. 45. 5 to 4 cm;. LIGHT For Exercises 45 and 46, use the table below. 45. Express the range of the wavelengths of visible light using positive exponents. Then evaluate each expression. 46. Express the range of the wavelengths of X-rays using positive exponents. Then evaluate each expression.. c will decrease.. Reading the Lesson am 1. Explain what the statement am n means. an pH. , you can find the power of the numerator, the power of. the denominator, and divide. This is an example of what property?. 422. Chapter 8 Polynomials. Power of a Quotient Property 3. Use the Quotient of Powers Property to explain why 30 1. Sample answer:. 34 1. The Quotient of Powers Property says that when you divide 34. two powers that have the same base, you subtract the exponents.. NAME ______________________________________________ DATE. 4. 3 0 So 4 3 .. 8-2 Enrichment Enrichment,. 3. 4. Consider the expression 43.. ____________ PERIOD _____. p. 466. a. Explain why the expression 43 is not simplified. An expression involving. exponents is not considered simplified if the expression contains negative exponents. b. Define the term reciprocal. The reciprocal of a number is 1 divided by. the number.. Patterns with Powers Use your calculator, if necessary, to complete each pattern. a. 210 . c. 43 is the reciprocal of what power of 4? 43. 1 4. 29 . 1 64. d. What is the simplified form of 43? or 3. 28. . 27 . Helping You Remember 4x2 2x. 5. Describe how you would help a friend who needs to simplify the expression 5 .. Divide the constants and group powers with the same base to get. 26 25 . 2. 24 . 5. 23. x . Use the Quotient of Powers Property to get (2)(x 25) or (2)(x3 ). 42 x . To simplify (2)(x3), use the Negative Exponent Property to get. . 1 2 (2) , or . x3 x3. . 22 21. . 1024 512 256 128 64 32 16 8 4 2. b. 510 59 58. . 57 56 55 54 53. . 52 51. . 9,765,625 1,953,125 390,625 78,125 15,625 3125 625 125 25 5. c. 410 49 48. . 47 46 45 44 43 42 41 . Study the patterns for a, b, and c above. Then answer the questions.. 422. Chapter 8 Polynomials. Spectrum of Electromagnetic Radiation Region Radio Microwave. 1,048,576 262,144 65,536 16,384 4096 1024 256 64 16 4. Wavelength (cm) greater than 10 101 to 102. Infrared. 102 to 105. Visible. 105 to 104. Ultraviolet. 104 to 107. X-rays. 107 to 109. Gamma Rays. To divide two powers that have the same base, subtract the exponents.. 101 . 1 2. 43. Write an expression to represent the probability of tossing a coin n times and getting n heads.. Source: www.school.discovery.com. • How do you think c will change as the exponent increases?. 2. To find c in the formula c . 1 2 2. the probability of getting heads each time is or .. less than 109.

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