Division Properties of Exponents

Page 2 of 7EXAMPLE 2Using the Power of a Quotient PropertySTUDENT HELPHOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor extra examples.INTERNETSimplify the expression.a. 2 3 2 b.º 3 y 3 c. 7 4 º3SOLUTIONa. 2 3 2 = 2 32 = 4 9 2Square numerator and denominator and simplify.b.º 3 y 3 = º 3y 3= (º 33y) 3= º 273yc. 7 4 º3 = 7 4333= 4 7364= 3 43Rewrite fraction.Power of a quotientSimplify.Power of a quotientDefinition of negative exponentsSimplify.EXAMPLE 3Simplifying ExpressionsSimplify the expression. 42xa. • b. 2 y 9xy 22 x 3x y42ySOLUTION2x 2 y 9xy (2x 2 y)(9xy 2 )a. • 2 = Multiply fractions.3x y4(3x)(y4 )18x= y 3 3xy4Product of powers= 6x 2 y º1 Quotient of powers2xb. 4 = y2= 6 x2 y(2x) 4(y2 ) 4Definition of negative exponentsPower of a quotient2= 4 • x 4Product of a product and power of a powery2• 416x= 4Simplify.y8464 Chapter 8 Exponents and Exponential Functions

Page 3 of 7FOCUS ONCAREERSGOAL 2USING POWERS AS REAL-LIFE MODELSEXAMPLE 4Using the Quotient of Powers PropertySTOCK EXCHANGE The number of shares N (in billions) listed on the New YorkStock Exchange from 1977 through 1997 can be modeled byN = 92.56 • (1.112) twhere t = 0 represents 1990. Find the ratio of shares listed in 1997 to the shareslisted in 1977. Source: Federal Reserve Bank of New YorkFLOOR TRADERFloor traders at theNew York Stock Exchangeuse hand signals that dateback to the 1880s to relayinformation about theirtrades.CAREER LINKwww.mcdougallittell.comREALINTERNETLIFESOLUTIONUse t = º13 for 1977 and t = 7 for 1997.Number listed in 1997Number listed in 1977=92.56 • (1.112) 792.56 • (1.112)º13= 1.112 7 º (º13)= 1.112 20≈ 8.4 The ratio of shares listed in 1997 to the shares listed in 1977 is 8.4 to 1.There were about 8.4 times as many listed in 1997 as in 1977.EXAMPLE 5Using the Power of a Quotient PropertyPROBABILITY CONNECTION You toss a fair coin ten times. Show that theprobability that the coin lands heads up each time is about 0.001.SOLUTIONProbability that the first toss is heads:Probability that the first two tosses are heads: 1 2 2Probability that the first three tosses are heads: 1 2 3••Probability that the first nine tosses are heads: 1 2 9Probability that the first ten tosses are heads: 1 2 10 1 2 STUDENT HELPLook BackFor help with probability,see page 114.Use the power of a quotient property to evaluate. 1 2 10 1= 210 = 1 ≈ 0.0011024 The probability is 1 2 10 or about 0.001.8.3 Division Properties of Exponents 465

Page 4 of 7GUIDED PRACTICEVocabulary Check ✓Concept Check ✓Skill Check ✓41. The expression a a6 can be simplified by using the ? property.x 82. Can be simplified? Explain your answer.y3Use the quotient of powers property to simplify the expression.3. 5 451 4. 7 65a m79 5. 126. a9m117. a 5 8. ( 8º 2) 9. 53 • 5 5x 10.7 • x239a( º 2)5xº2Use the power of a quotient property to simplify the expression.11. 1 2 5 12. 3 5 3 13. m5 2 14. 2 b 415. 5 4 º3 x16. 4 23 2 17. x 3 6 y 5 18. a 6b9 5PRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 804.EVALUATING EXPRESSIONS Evaluate the expression. Write fractions insimplest form.19. 5 653 20. 8 381 21. (º 693)( º 3) 22. 6º 3(9º 3)3323. 3º 4 24. 83 •58 25. 5 •5 26. 3 84 28 227. 6 2 3 28.º 2 3 3 29.º 3 5 2 30. 9 6 º15 4SIMPLIFYING EXPRESSIONS Simplify the expression. The simplifiedexpression should have no negative exponents.31. 4 32. x 431 33. 5x 5 34. x 3 1• x xa8 36. 37. º1 º2 m38.3 • m 9 5a5y3m239. 40. 41. 3 3 3(r 3 ) 4º6x 2 y2x 3 y 416xy 42. (r3 ) 8 º35. x 5 • x12xy3 xy 2 3xy22xy • º 34xy–– 1xSTUDENT HELPHOMEWORK HELPExample 1: Exs. 19–50Example 2: Exs. 19–48Example 3: Exs. 31–50Example 4: Exs. 51–59Example 5: Exs. 60, 613 3y43. 4 x2x y• 5 x2 y5º8y 28 b 244. 36 a 6 •aba º1b 23 2 4 6x º2 y 246. • xyº345. 16 x7x y4y• x y(4x 2 y) 8 xy º2xy25x 2xy47. • 48. º3 y 2 (2xy 3 ) º2º2 y 4 º2 4xy • xyx 5 3x º1 2xº 1y º1 y 2y º3466 Chapter 8 Exponents and Exponential Functions

Page 5 of 7ERROR ANALYSIS In Exercises 49 and 50, find and correct the errors.349.6 3 ÷ 6= 6 50.6=1 3=1– 9 = xx––9 – 3x3= x –121= x– 1251. EARTH AND MOON The volumeof a sphere is given by V = 4 3 πr3 , wherer is the radius of the sphere. Assumingthat the radius of the Moon is 1 4 the radiusof Earth, find the ratio of the volume ofEarth to the volume of the Moon. Let rrepresent the radius of Earth.Not drawn to scaleMoonEarthr52. RETAIL SALES From 1994 to 1998,the sales for a national clothing storeincreased by about the same percenteach year. The sales S (in millions ofdollars) for year t can be modeled byS = 3723 6 5 twhere t = 0 corresponds to 1994. Findthe ratio of 1998 sales to 1995 sales.Sales(millions of dollars)S70006000500040003000200010000 01 2 3 4 tYears since 1994ATLANTIC CODAdult Atlantic codaverage about 3 feet inlength and weigh from 10 to25 pounds, though some maygrow much larger.REALFOCUS ONAPPLICATIONSLIFE53. BASEBALL SALARIES The average salary for a professional baseballplayer in the United States can be approximated by y = 283(1.2) t , wheret = 0 represents the year 1984. Using this approximation, find the ratio of anaverage salary in 1988 to an average salary in 1994.ATLANTIC COD In Exercises 54–56, use the following information.The average weight w (in pounds) of an Atlantic cod t years old can bemodeled by the equation w = 1.16(1.44) t . Source: National Marine Fisheries Service,Northeast Science Center54. Find the ratio of the weight of a 5-year-old cod to the weight of a 2-year-oldcod. Express this ratio as a power of 1.44.55. A 5-year-old cod weighs how many times as much as a 2-year-old cod?56. According to the model, what is the average weight of an Atlantic cod whenit is hatched? How did you get your answer?57. SALES From 1995 through 1999, the sales for a national furniture storeincreased by about the same percent each year. The sales s (in millions ofdollars) for year t can be modeled by s = 476(1.13) t , where t = 0 represents1995. Find the ratio of 1997 sales to 1999 sales.8.3 Division Properties of Exponents 467

Page 6 of 7ROWING Shellswith 4 or 8 rowersusually have an additionalnonrowing member of theteam to direct the rest of thecrew. This person is calledthe coxswain.REALFOCUS ONAPPLICATIONSLIFETestPreparation★ ChallengeEXTRA CHALLENGEwww.mcdougallittell.com58. OLYMPIC ROWING The racing shells (boats) used in rowing competitionusually have 1, 2, 4, or 8 rowers. Top speeds for racing shells in theOlympic 2000-meter races can be modeled by s = 16.3(1.0285) n , where sis the speed in kilometers per hour and n is the number of rowers. Use themodel to estimate the ratio of the speed of an 8-rower shell to the speed ofa 2-rower shell.59. LEARNING SPANISH You memorized a list of 200 Spanish vocabularywords. Unfortunately, each week you forget one fifth of the words you knewthe previous week. The number of words S you remember after n weeks canbe approximated by the following equation.Vocabulary words remembered: S = 200 4 5 na. Copy and complete the table showing the number of words you rememberafter n weeks.b. CRITICAL THINKING How many weeks does it take to forget all butthree words? Explain your answer.60. PROBABILITY CONNECTION You roll a die eight times. What is the probabilitythat you will roll eight sixes in a row?61. PROBABILITY CONNECTION You roll a die six times. What is the probabilitythat you will roll six even numbers in a row?MULTI–STEP PROBLEM In Exercises 62–65, use the following information.You work for a real estate company that wants to build a new apartment complex.A team is formed to decide in which state to build the complex. One teammember wants to build in Arizona. Another team member wants to build inMichigan. Your boss asks you to decide where to build the complex.62. You find that the population P of Arizona (in thousands) in 1995 projectedthrough 2025 can be modeled by P = 4264(1.0208) t , where t = 0 represents1995. Find the ratio of the population in 2025 to the population in 2000.INTERNETWeeks, n 0 1 2 3 4 5 6Words, S ? ? ? ? ? ? ?DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com63. You find that the population P of Michigan (in thousands) in 1995 projectedthrough 2025 can be modeled by P = 9540(1.0026) t , where t = 0 represents1995. Find the ratio of the population in 2025 to the population in 2000. Source: U.S. Bureau of the Census64. Which population is projected to grow more rapidly?65. Writing Use the results from Exercises 62–64 to decide where to build thecomplex. Write a memo to your boss explaining your decision.66. STACKING PAPER A piece of notebook paper is about 0.0032 inchthick. If you begin with a stack consisting of a single sheet and double thestack 25 times, how tall will the stack be in inches? How tall will it be infeet? (Hint: Write and solve an exponential equation to find the height of thestack in inches. Then use unit analysis to find the height in feet.)468 Chapter 8 Exponents and Exponential Functions

Page 7 of 7MIXED REVIEWPOWERS OF TEN Evaluate the expression. (Review 1.2, 8.2 for 8.4)67. 10 5 68. 10 3 69. 10 º4 70. 10 º8SKETCHING GRAPHSplane. (Review 6.5)Sketch the graph of the inequality in a coordinate71. x ≥ 5 72. x + 3 < 4 73. y > º2 74. y ≤ º1.575. x ≥ 2.5 76. 3x º y < 0x77. y ≤ 23 78. 4 x + 1 4 y ≥ 1CHECKING FOR SOLUTIONSof the system. (Review 7.1)Decide whether the ordered pair is a solution79. 2x + 4y = 2 80. x º 5y = 9ºx + 5y = 13 (º3, 2) 3x + y = 11 (1, º4)81. 8a + 4b = 6 82. 3c º 8d = 114a + b = 3 3 4 , 0 c + 6d = 8 5, º 1 2 SOLVING LINEAR SYSTEMS(Review 7.3)Use linear combinations to solve the system.83. x º y = 4 84. ºx + 2y = 12 85. 2a + 3b = 17x + y = 12 x + 6y = 20 3a + 4b = 24QUIZ 1 Self-Test for Lessons 8.1–8.3Evaluate the expression. (Lessons 8.1, 8.2, and 8.3)1. 3 3 • 3 4 2. (2 2 ) 4 3. (8 + 2) 2 2 4. 7 º45. 4 º3 • 4 º4 6. 6 7 º1 7. 5 8.259. 5 4 º3 (º2) 910. 11. 6 0 •1 12. 23 º4• 2 (º2)243º 32º 33 4 • 3 6 33Simplify the expression. Write your answer with no negative exponents.(Lessons 8.1, 8.2, and 8.3)13. x 4 • x 5 14. (º2x) 5 15. ºa3º5 16. 2000 c 5617. x 4xº 518. x º2mº 619. 2 n 4 20. x 4 1 • x3mn2x320x 3 y º6xy 4xy2 ºx21. (3a) 3 • (º4a) 3 22. 8m 3 2 1 2 m2 2 23. •24. SAVINGS ACCOUNT You started a savings account in 1994. The balanceA is given by A = 250(1.08) t , where t = 0 represents the year 2001. What isthe balance in the account in 1994? in 1999? in 2001? (Lesson 8.2)8.3 Division Properties of Exponents 469