Exponents and Exponential Functions. Prerequisite Skills. Before VOCABULARY CHECK SKILLS CHECK

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1 Eponents and Eponential Functions 88. Appl Eponent Properties Involving Products 8. Appl Eponent Properties Involving Quotients 8.3 Define and Use Zero and Negative Eponents 8.4 Use Scientific Notation 8.5 Write and Graph Eponential Growth Functions 8.6 Write and Graph Eponential Deca Functions Before In previous chapters and courses, ou learned the following skills, which ou ll use in Chapter 8: evaluating epressions involving eponents, ordering numbers, writing percents as decimals, and writing function rules. Prerequisite Skills VOCABULARY CHECK. Identif the eponent and the base in the epression Cop and complete: An epression that represents repeated multiplication of the same factor is called a(n)?. SKILLS CHECK Evaluate the epression. (Review p. for ) 3. when a 3 when a r when r z 3 when z 5 Order the numbers from least to greatest. (Review p. 909 for 8.4.) 7. 6., 6., , 0.0, Write the percent as a decimal. (Review p. 96 for 8.5 and 8.6.) 9. 4% %. 3.8%. 45% 3. Write a rule for the function. (Review p. 35 for 8.5 and 8.6.) Input Output

2 Now In Chapter 8, ou will appl the big ideas listed below and reviewed in the Chapter Summar on page 54. You will also use the ke vocabular listed below. Big Ideas Appling properties of eponents to simplif epressions Working with numbers in scientific notation 3 Writing and graphing eponential functions KEY VOCABULARY order of magnitude, p. 49 scientific notation, p. 5 eponential function, p. 50 eponential growth, p. 5 compound interest, p. 53 eponential deca, p. 533 Wh? You can use eponents to eplore eponential growth and deca. For eample, ou can write an eponential function to find the value of a collector car over time. Algebra The animation illustrated below for Eample 4 on page 5 helps ou answer this question: If ou know the growth rate of the value of a collector car over time, can ou predict what the car will sell for at an auction? Find the value of the collector car over time. Algebra at www. publisher.com Click on the boes to enter the initial value and growth rate. Algebra at classzone.com Other animations for Chapter 8: pages 49, 505, 5, 534, and

3 Investigating Algebra ACTIVITY Use before Lesson Products and Powers MATERIALS paper and pencil QUESTION E XPLORE How can ou find a product of powers and a power of a power? Find products of powers STEP Cop and complete Cop and complete the table. Epression Epression as repeated multiplication Number of factors Simplified epression 7 4 p 7 5 (7 p 7 p 7 p 7) p (7 p 7 p 7 p 7 p 7) (4) p (4) 3 [(4) p (4)] p [(4) p (4) p (4)]?? p 5??? STEP Analze results Find a pattern that relates the eponents of the factors in the first column and the eponent of the epression in the last column. E XPLORE Find powers of powers STEP Cop and complete Cop and complete the table. Epression Epanded epression Epression as repeated multiplication Number of factors Simplified epression (5 3 ) (5 3 ) p (5 3 ) (5 p 5 p 5) p (5 p 5 p 5) [(6) ] 4 [(6) ] p [(6) ] p [(6) ] p [(6) ]??? (a 3 ) 3???? STEP Analze results Find a pattern that relates the eponents of the epression in the first column and the eponent of the epression in the last column. DRAW CONCLUSIONS Use our observations to complete these eercises Simplif the epression. Write our answer using eponents.. 5 p 5 3. (6) p (6) 4 3. m 6 p m 4 4. (0 3 ) 3 5. [() 3 ] 4 6. (c ) 6 In Eercises 7 and 8, cop and complete the statement. 7. If a is a real number and m and n are positive integers, then a m p a n 5?. 8. If a is a real number and m and n are positive integers, then (a m ) n 5?. 488 Chapter 8 Eponents and Eponential Functions

4 8. Appl Eponent Properties Involving Products Before You evaluated eponential epressions. Now You will use properties of eponents involving products. Wh? So ou can evaluate agricultural data, as in Eample 5. Ke Vocabular order of magnitude power, p. 3 eponent, p. 3 base, p. 3 Notice what happens when ou multipl two powers that have the same base. 5 factors a p a 3 5 (a p a) p (a p a p a) 5 a 5 5 a 3 factors 3 factors The eample above suggests the following propert of eponents, known as the product of powers propert. KEY CONCEPT For Your Notebook Product of Powers Propert Let a be a real number, and let m and n be positive integers. Words To multipl powers having the same base, add the eponents. Algebra a m p a n 5 a m n Eample 5 6 p SIMPLIFY EXPRESSIONS When simplifing powers with numerical bases onl, write our answers using eponents, as in parts (a), (b), and (c). E XAMPLE Use the product of powers propert a. 7 3 p b. 9 p 9 8 p p 9 8 p c. (5)(5) 6 5 (5) p (5) 6 5 (5) 6 5 (5) 7 d. 4 p GUIDED PRACTICE for Eample Simplif the epression.. 3 p p (7) (7) 4. p 6 p 8. Appl Eponent Properties Involving Products 489

5 POWER OF A POWER Notice what happens when ou raise a power to a power. (a ) 3 5 a p a p a 5 (a p a) p (a p a) p (a p a) 5 a 6 5 a p 3 The eample above suggests the following propert of eponents, known as the power of a power propert. KEY CONCEPT For Your Notebook Power of a Power Propert Let a be a real number, and let m and n be positive integers. Words To find a power of a power, multipl eponents. Algebra (a m ) n 5 a mn Eample (3 4 ) p E XAMPLE Use the power of a power propert AVOID ERRORS In part (d), notice that ou can write [( ) 6 ] as ( ), but ou cannot write ( ) as. a. ( 5 ) p 3 b. [(6) ] 5 5 (6) p (6) 0 c. ( ) 4 5 p 4 d. [( ) 6 ] 5 ( ) 6 p ( ) GUIDED PRACTICE for Eample Simplif the epression. 5. (4 ) 7 6. [() 4 ] 5 7. (n 3 ) 6 8. [(m ) 5 ] 4 POWER OF A PRODUCT Notice what happens when ou raise a product to a power. (ab) 3 5 (ab) p (ab) p (ab) 5 (a p a p a) p (b p b p b) 5 a 3 b 3 The eample above suggests the following propert of eponents, known as the power of a product propert. KEY CONCEPT For Your Notebook Power of a Product Propert Let a and b be real numbers, and let m be a positive integer. Words To find a power of a product, find the power of each factor and multipl. Algebra (ab) m 5 a m b m Eample (3 p 7) p Chapter 8 Eponents and Eponential Functions

6 SIMPLIFY EXPRESSIONS When simplifing powers with numerical and variable bases, be sure to evaluate the numerical power, as in parts (b), (c), and (d). E XAMPLE 3 Use the power of a product propert a. (4 p 3) p 3 8 b. (9) 5 (9 p p ) 5 9 p p 5 8 c. (4z) 5 (4 p z) 5 (4) p z 5 6z d. (4z) 5(4 p z) 5(4 p z ) 56z E XAMPLE 4 Use all three properties Simplif ( 3 ) p 4. ( 3 ) p 4 5 p ( 3 ) p 4 Power of a product propert 5 4 p 6 p 4 Power of a power propert Product of powers propert at classzone.com ORDER OF MAGNITUDE The order of magnitude of a quantit can be defined as the power of 0 nearest the quantit. Order of magnitude can be used to estimate or perform rough calculations. For instance, there are about 9,000 species of insects in the United States. The power of 0 closest to 9,000 is 0 5, or 00,000. So, there are about 0 5 species of insects in the United States. E XAMPLE 5 Solve a real-world problem BEES In 003 the U.S. Department of Agriculture (USDA) collected data on about 0 3 honebee colonies. There are about 0 4 bees in an average colon during hone production season. About how man bees were in the USDA stud? Solution To find the total number of bees, find the product of the number of colonies, 0 3, and the number of bees per colon, p c The USDA studied about 0 7, or 0,000,000, bees. p6 p6 GUIDED PRACTICE for Eamples 3, 4, and 5 Simplif the epression. 9. (4 p ) 0. (3n). (9m 3 n) 4. 5 p (5 ) 4 3. WHAT IF? In Eample 5, 0 honebee colonies in the stud were located in Idaho. About how man bees were studied in Idaho? 8. Appl Eponent Properties Involving Products 49

7 8. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3 and 55 5 STANDARDIZED TEST PRACTICE Es., 40, 4, 50, and 58 5 MULTIPLE REPRESENTATIONS E. 55. VOCABULARY Cop and complete: The? of the quantit 93,534,004 people is the power of 0 nearest the quantit, or 0 8 people.. WRITING Eplain when and how to use the product of powers propert. EXAMPLES,,3, and 4 on pp for Es. 3 4 SIMPLIFYING EXPRESSIONS Simplif the epression. Write our answer using eponents p p p p (7) 4 (7) 5 8. (6) 6 (6) 9. 4 p 9 p 0. (3) (3) (3). (3 5 ). (7 4 ) 3 3. [(5) 3 ] 4 4. [(8) 9 ] 5. (5 p 9) 3 6. (7 p 6) 4 7. (3 p 9) 6 8. ((4) p ) 5 SIMPLIFYING EXPRESSIONS Simplif the epression p 0. 9 p. z p z p z 3. a 4 p a 3 p a 0 3. ( 5 ) 4. ( 4 ) 6 5. [(b ) ] 6 6. [(d 9) 7 ] 3 7. (5) 8. (5) 9. (7) 30. (5pq) 3 3. (0 6 ) p 3. (8m 4 ) p m d p (d 5 ) (0 3 ) ( 7 ) 35. (p 4 ) 3 (.5p 7 ) ( ) (3 5 ) 3 ( 7 ) 38. (0n) (4n 3 ) ERROR ANALYSIS Describe and correct the error in simplifing c p c 4 p c 5. c p c 4 p c 5 5 c p c 4 p c 5 5c p 4 p 5 5c MULTIPLE CHOICE Which epression is equivalent to (9) 6? A (9) (9) 3 B (9)(9) 5 C [(9) 4 ] D [(9) 3 ] 3 4. MULTIPLE CHOICE Which epression is equivalent to 36? A (6 3 ) 4 B 4 p 3 3 C 3 3 p (4 3 ) 3 D (6 5 ) p SIMPLIFYING EXPRESSIONS Find the missing eponent p? ( 8 )? (z? ) 3 5 8z (3a 3 )? p a 3 5 8a POPULATION The population of New York Cit in 000 was 8,008,78. What was the order of magnitude of the population of New York Cit? SIMPLIFYING EXPRESSIONS Simplif the epression. 47. (3 ) 3 ( 3 5 ) 48. ( z 3 ) 5 ( 4 z) 49. (s)(5r 3 st) 3 (r 4 st 7 ) 49 Chapter 8 Eponents and Eponential Functions

8 50. OPEN ENDED Write three epressions involving products of powers, powers of powers, or powers of products that are equivalent to CHALLENGE Show that when a and b are real numbers and n is a positive integer, (ab) n 5 a n b n. PROBLEM SOLVING EXAMPLE 5 on p. 49 for Es ICE CREAM COMPOSITION There are about 954,930 air bubbles in cubic centimeter of ice cream. There are about 946 cubic centimeters in quart. Use order of magnitude to find the approimate number of air bubbles in quart of ice cream. 53. ASTRONOMY The order of magnitude of the radius of our solar sstem is 0 3 meters. The order of magnitude of the radius of the visible universe is 0 3 times as great. Find the approimate radius of the visible universe. 54. COASTAL LANDSLIDE There are about billion grains of sand in cubic foot of sand. In 995 a stretch of beach at Sleeping Bear Dunes National Lakeshore in Michigan slid into Lake Michigan. Scientists believe that around 35 million cubic feet of sand fell into the lake. Use order of magnitude to find about how man grains of sand slid into the lake. 55. MULTIPLE REPRESENTATIONS There are about 0 3 atoms of gold in ounce of gold. a. Making a Table Cop and complete the table b finding the number of atoms of gold for the given amounts of gold (in ounces). Gold (ounces) ,000 00,000 Number of atoms????? b. Writing an Epression A particular mine in California etracted about 96,000 ounces of gold in ear. Use order of magnitude to write an epression ou can use to find the approimate number of atoms of gold etracted in the mine that ear. Simplif the epression. Verif our answer using the table. 56. MULTI-STEP PROBLEM A microscope has two lenses, the objective lens and the eepiece, that work together to magnif an object. The total magnification of the microscope is the product of the magnification of the objective lens and the magnification of the eepiece. a. Your microscope s objective lens magnifies an object 0 times, and the eepiece magnifies an object 0 times. What is the total magnification of our microscope? b. You magnif an object that is 0 nanometers long. How long is the magnified image? Eepiece Objective lens 8. Appl Eponent Properties Involving Products 493

9 57. VOLUME OF THE SUN The radius of the sun is about 695,000,000 meters. The formula for the volume of a sphere, such as the sun, is V 5 4 πr 3. 3 Because the order of magnitude of 4 π is, it does not contribute to the 3 formula in a significant wa. So, ou can find the order of magnitude of the volume of the sun b cubing its radius. Find the order of magnitude of the volume of the sun. 58. EXTENDED RESPONSE Rock salt can be mined from large deposits of salt called salt domes. A particular salt dome is roughl clindrical in shape. The order of magnitude of the radius of the salt dome is 0 3 feet. The order of magnitude of the height of the salt dome is about 0 times that of its radius. The formula for the volume of a clinder is V 5 πr h. a. Calculate What is the order of magnitude of the height of the salt dome? b. Calculate What is the order of magnitude of the volume of the salt dome? c. Eplain The order of magnitude of the radius of a salt dome can be 0 times the radius of the salt dome described in this eercise. What effect does multipling the order of magnitude of the radius of the salt dome b 0 have on the volume of the salt dome? Eplain. 59. CHALLENGE Your school is conducting a poll that has two parts, one part that has 3 questions and a second part that has 0 questions. Students can answer the questions in either part with agree or disagree. What power of represents the number of was there are to answer the questions in the first part of the poll? What power of represents the number of was there are to answer the questions in the second part of the poll? What power of represents the number of was there are to answer all of the questions on the poll? Salt MIXED REVIEW PREVIEW Prepare for Lesson 8. in Es Find the product. (p. 88) Evaluate the epression for the given value of the variable. (p. ) when when when Graph the equation or inequalit (p. 5) (p. 5) (p. 5) (p. 44) (p. 44) 7. 3 (p. 405) 7. <.5 (p. 405) (p. 405) 74. < 3 (p. 405) 494 EXTRA PRACTICE for Lesson 8., p. 945 ONLINE QUIZ at classzone.com

10 8. Appl Eponent Properties Involving Quotients Before You used properties of eponents involving products. Now You will use properties of eponents involving quotients. Wh? So ou can compare magnitudes of earthquakes, as in E. 53. Ke Vocabular power, p. 3 eponent, p. 3 base, p. 3 Notice what happens when ou divide powers with the same base. a 5 5 a p a p a p a p a a 3 a p a p a 5 a p a 5 a 5 a 5 3 The eample above suggests the following propert of eponents, known as the quotient of powers propert. KEY CONCEPT For Your Notebook Quotient of Powers Propert Let a be a nonzero real number, and let m and n be positive integers such that m > n. Words To divide powers having the same base, subtract eponents. Algebra am a n 5 am n, a? 0 Eample E XAMPLE Use the quotient of powers propert SIMPLIFY EXPRESSIONS When simplifing powers with numerical bases onl, write our answers using eponents, as in parts (a), (b), and (c). a. c b. 5 4 p (3) 9 (3) 3 5 (3) (3) d. 4 p GUIDED PRACTICE for Eample Simplif the epression (4) 9 (4) p p 8 8. Appl Eponent Properties Involving Quotients 495

11 POWER OF A QUOTIENT Notice what happens when ou raise a quotient to a power. a b 4 5 a p a p a p a 5 a p a p a p a 5 a4 b b b b b p b p b p b b 4 The eample above suggests the following propert of eponents, known as the power of a quotient propert. KEY CONCEPT For Your Notebook Power of a Quotient Propert Let a and b be real numbers with b Þ 0, and let m be a positive integer. Words To find a power of a quotient, find the power of the numerator and the power of the denominator and divide. Algebra a b m 5 am b m, b Þ 0 Eample SIMPLIFY EXPRESSIONS When simplifing powers with numerical and variable bases, evaluate the numerical power, as in part (b). E XAMPLE a E XAMPLE 3 a (4 ) 3 (5) 3 Use the power of a quotient propert Use properties of eponents Power of a quotient propert b (7) p ( ) Power of a product propert Power of a power propert b. a b 5 p 5 (a ) 5 p a b 5 a Power of a quotient propert 5 a0 b 5 p a Power of a power propert 5 a0 a b 5 Multipl fractions. 5 a8 b 5 Quotient of powers propert 496 Chapter 8 Eponents and Eponential Functions

12 GUIDED PRACTICE for Eamples and 3 Simplif the epression. 5. a b s 3t 3 p t5 6 E XAMPLE 4 Solve a multi-step problem FRACTAL TREE To construct what is known as a fractal tree, begin with a single segment (the trunk) that is unit long, as in Step 0. Add three shorter segments that are unit long to form the first set of branches, as in Step. Then continue adding sets of successivel shorter branches so that each new set of branches is half the length of the previous set, as in Steps and 3. Step 0 Step Step Step 3 a. Make a table showing the number of new branches at each step for Steps 4. Write the number of new branches as a power of 3. b. How man times greater is the number of new branches added at Step 5 than the number of new branches added at Step? Solution a. Step Number of new branches b. The number of new branches added at Step 5 is 3 5. The number of new branches added at Step is 3. So, the number of new branches added at Step 5 is times the number of new branches added at Step. GUIDED PRACTICE for Eample 4 9. FRACTAL TREE In Eample 4, add a column to the table for the length of the new branches at each step. Write the lengths of the new branches as powers of. What is the length of a new branch added at Step 9? 8. Appl Eponent Properties Involving Quotients 497

13 E XAMPLE 5 Solve a real-world problem ASTRONOMY The luminosit (in watts) of a star is the total amount of energ emitted from the star per unit of time. The order of magnitude of the luminosit of the sun is 0 6 watts. The star Canopus is one of the brightest stars in the sk. The order of magnitude of the luminosit of Canopus is 0 30 watts. How man times more luminous is Canopus than the sun? Solution Luminosit of Canopus (watts) Luminosit of the sun (watts) Canopus c Canopus is about 0 4 times as luminous as the sun. GUIDED PRACTICE for Eample 5 0. WHAT IF? Sirius is considered the brightest star in the sk. Sirius is less luminous than Canopus, but Sirius appears to be brighter because it is much closer to Earth. The order of magnitude of the luminosit of Sirius is 0 8 watts. How man times more luminous is Canopus than Sirius? 8. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 33 and 5 5 STANDARDIZED TEST PRACTICE Es., 9, 37, 46, and 54 5 MULTIPLE REPRESENTATIONS E. 49. VOCABULARY Cop and complete: In the power 4 3, 4 is the? and 3 is the?.. WRITING Eplain when and how to use the quotient of powers propert. EXAMPLES and on pp for Es. 30 SIMPLIFYING EXPRESSIONS Simplif the epression. Write our answer using eponents (6) 8 (6) 5 7. (4) 7 (4) 4 8. () 9 () p p p 7 6. p p p Chapter 8 Eponents and Eponential Functions

14 9. MULTIPLE CHOICE Which epression is equivalent to 6 6? A B 6 6 C D ERROR ANALYSIS Describe and correct the error in simplifing 95 p p EXAMPLES,, and 3 on pp for Es. 37 SIMPLIFYING EXPRESSIONS Simplif the epression.. 8 p 5. z 8 p z 7 3. a 9 4. j k 5. p q a b c d a 7 b p p p 8m 5 m4 n p MULTIPLE CHOICE Which epression is equivalent to 73 4? A B C D SIMPLIFYING EXPRESSIONS Find the missing eponent. 38. (8) 7 5 (8)? (8) ? p p p 5 p? 5 p 9 4. c 3 d? 5 6c d 8 SIMPLIFYING EXPRESSIONS Simplif the epression. 4. f g 3 3fg s3 t 3 st p (3st)3 s t 44. m 5 n 4m p mn4 5n p OPEN ENDED Write three epressions involving quotients that are equivalent to REASONING Name the definition or propert that justifies each step to show that am a n 5 a n m for m < n. Let m < n. Given a m a n 5 am a n a m a m? 5 a n a m? 5 a n m? 48. CHALLENGE Find the values of and if ou know that b b 5 b9 and b p b 5 b 3. Eplain how ou found our answer. b 3 8. Appl Eponent Properties Involving Quotients 499

15 PROBLEM SOLVING EXAMPLES 4 and 5 on pp for Es MULTIPLE REPRESENTATIONS Draw a square with side lengths that are unit long. Divide it into four new squares with side lengths that are one half the side length of the original square, as shown in Step. Keep dividing the squares into new squares, as shown in Steps and 3. Step 0 Step Step Step 3 a. Making a Table Make a table showing the number of new squares and the side length of a new square at each step for Steps 4. Write the number of new squares as a power of 4. Write the side length of a new square as a power of. b. Writing an Epression Write and simplif an epression to find b how man times the number of squares increased from Step to Step GROSS DOMESTIC PRODUCT In 003 the gross domestic product (GDP) for the United States was about trillion dollars, and the order of magnitude of the population of the U.S. was 0 8. Use order of magnitude to find the approimate per capita (per person) GDP? 5. SPACE TRAVEL Alpha Centauri is the closest star sstem to Earth. Alpha Centauri is about 0 3 kilometers awa from Earth. A spacecraft leaves Earth and travels at an average speed of 0 4 meters per second. About how man ears would it take the spacecraft to reach Alpha Centauri? 5. ASTRONOMY The brightness of one star relative to another star can be measured b comparing the magnitudes of the stars. For ever increase in magnitude of, the relative brightness is diminished b a factor of.5. For instance, a star of magnitude 8 is.5 times less bright than a star of magnitude 7. The constellation Ursa Minor (the Little Dipper) is shown. How man times less bright is Eta Ursae Minoris than Polaris? Ursa Minor Eta Ursae Minoris (magnitude 5) Polaris (magnitude ) 53. EARTHQUAKES The energ released b one earthquake relative to another earthquake can be measured b comparing the magnitudes (as determined b the Richter scale) of the earthquakes. For ever increase of in magnitude, the energ released is multiplied b a factor of about 3. How man times greater is the energ released b an earthquake of magnitude 7 than the energ released b an earthquake of magnitude 4? 5 WORKED-OUT SOLUTIONS 500 Chapter 8 Eponents p. WSand Eponential Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

16 54. EXTENDED RESPONSE A bte is a unit used to measure computer memor. Other units are based on the number of btes the represent. The table shows the number of btes in certain units. For eample, from the table ou can calculate that terabte is equivalent to 0 gigabtes. a. Calculate How man kilobtes are there in terabte? Unit Number of btes b. Calculate How man megabtes are there in Kilobte 0 petabte? Megabte 0 c. CHALLENGE Another unit used to measure computer memor is a bit. There are 8 bits Gigabte 30 in a bte. Eplain how ou can convert the Terabte 40 number of btes per unit given in the table to the number of bits per unit. Petabte 50 MIXED REVIEW PREVIEW Prepare for Lesson 8.3 in Es Solve the equation. Check our solution. (p. 34) k t v z z Write an equation of the line that passes through the given points. (p. 9) 6. (, ), (0, 5) 6. (0, 3), (4, ) 63. (0, 3), (7, 3) 64. (4, 3), (5, 6) 65. (4, ), (, 4) 66. (, 3), (3, ) QUIZ for Lessons Simplif the epression. Write our answer using eponents.. 3 p 3 6 (p. 489). (5 4 ) 3 (p. 489) 3. (3 p 4) 7 (p. 489) 4. 7 p 7 6 p 7 (p. 489) 5. (4)(4) 9 (p. 489) (p. 495) (9) 9 (9) 7 (p. 495) p 3 4 (p. 495) (p. 495) Simplif the epression. 0. p 5 (p. 489). (3 3 ) (p. 489). (7) (p. 489) 3. (6 5 ) 3 p (p. 489) 4. ( 5 ) 3 (7 7 ) (p. 489) 5. 9 p (p. 495) (p. 495) 7. w v 6 (p. 495) (p. 495) 9. MAPLE SYRUP PRODUCTION In 00 the order of magnitude of the number of maple srup taps in Vermont was 0 6. The order of magnitude of the number of gallons of maple srup produced in Vermont was 0 5. About how man gallons of srup were produced per tap in Vermont in 00? (p. 495) EXTRA PRACTICE for Lesson 8., p. 945 ONLINE QUIZ at classzone.com 50

17 Investigating g Algebra ACTIVITY Use before Lesson Zero and Negative Eponents MATERIALS paper and pencil QUESTION E XPLORE How can ou simplif epressions with zero or negative eponents? Evaluate powers with zero and negative eponents STEP Find a pattern Cop and complete the tables for the powers of and 3. Eponent, n Value of n 4 6 3??? Eponent, n Value of 3 n 4 8 3??? As ou read the tables from the bottom up, ou see that each time the eponent is increased b, the value of the power is multiplied b the base. What can ou sa about the eponents and the values of the powers as ou read the table from the top down? STEP Etend the pattern Cop and complete the tables using the pattern ou observed in Step. Eponent, n Power, n Eponent, n Power, 3 n ???? 0? 0????? DRAW CONCLUSIONS Use our observations to complete these eercises. Find n and 3 n for n 53, 4, and 5.. What appears to be the value of a 0 for an nonzero number a? 3. Write each power in the tables above as a power with a positive eponent. For eample, ou can write 3 as Chapter 8 Eponents and Eponential Functions

18 8.3 Define and Use Zero and Negative Eponents Before You used properties of eponents to simplif epressions. Now You will use zero and negative eponents. Wh? So ou can compare masses, as in E. 5. Ke Vocabular reciprocal, p. 95 In the activit, ou saw what happens when ou raise a number to a zero or negative eponent. The activit suggests the following definitions. KEY CONCEPT For Your Notebook Definition of Zero and Negative Eponents Words Algebra Eample a to the zero power is. a 0 5, a Þ a n is the reciprocal of a n. a n is the reciprocal of a n. a n 5 a n, a Þ 0 a n 5 a n, a Þ E XAMPLE Use definition of zero and negative eponents SIMPLIFY EXPRESSIONS In this lesson, when simplifing powers with numerical bases, evaluate the numerical power. a Definition of negative eponents 5 9 Evaluate eponent. b. (7) 0 5 Definition of zero eponent c Definition of negative eponents 5 5 Evaluate eponent. 5 5 Simplif b multipling numerator and denominator b 5. d (Undefined) an is defined onl for a nonzero number a. GUIDED PRACTICE for Eample Evaluate the epression (8) () Define and Use Zero and Negative Eponents 503

19 PROPERTIES OF EXPONENTS The properties of eponents ou learned in Lessons 8. and 8. can be used with negative or zero eponents. KEY CONCEPT For Your Notebook Properties of Eponents Let a and b be real numbers, and let m and n be integers. a m p a n 5 a m n a m n 5 a mn ab m 5 a m b m Product of powers propert Power of a power propert Power of a product propert a m a n 5 am n, a Þ 0 Quotient of powers propert a b m 5 am b m, b Þ 0 Power of a quotient propert E XAMPLE Evaluate eponential epressions a. 6 4 p Product of powers propert Add eponents. 5 Definition of zero eponent b p Power of a power propert Multipl eponents Definition of negative eponents c Evaluate power Definition of negative eponents 58 Evaluate power. d Quotient of powers propert 55 3 Subtract eponents Definition of negative eponents 5 5 Evaluate power. GUIDED PRACTICE for Eample Evaluate the epression (5 3 ) 7. (3) 5 p (3) Chapter 8 Eponents and Eponential Functions

20 E XAMPLE 3 Use properties of eponents Simplif the epression. Write our answer using onl positive eponents. a. ( 5 ) p 3 p ( 5 ) 3 Power of a product propert 5 8 p 3 p 5 Power of a power propert b () () (4 ) 5 5 (4 )(4 ) Definition of negative eponents Definition of negative eponents Power of a product propert Product of powers propert Quotient of powers propert at classzone.com E XAMPLE 4 Standardized Test Practice The order of magnitude of the mass of a polphemus moth larva when it hatches is 0 3 gram. During the first 56 das of its life, the moth larva can eat about 0 5 times its own mass in food. About how man grams of food can the moth larva eat during its first 56 das? A 0 5 gram C 00 grams B gram D 0,000,000 grams Not to scale Solution To find the amount of food the moth larva can eat in the first 56 das of its life, multipl its original mass, 0 3, b p (3) The moth larva can eat about 00 grams of food in the first 56 das of its life. c The correct answer is C. A B C D GUIDED PRACTICE for Eamples 3 and 4 9. Simplif the epression 33. Write our answer using onl positive eponents SCIENCE The order of magnitude of the mass of a proton is 0 4 times greater than the order of magnitude of the mass of an electron, which is 0 7 gram. Find the order of magnitude of the mass of a proton. 8.3 Define and Use Zero and Negative Eponents 505

21 8.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. and 53 5 STANDARDIZED TEST PRACTICE Es., 44, 45, 54, and 57 5 MULTIPLE REPRESENTATIONS E. 55. VOCABULARY Which definitions or properties would ou use to simplif the epression 3 5 p 3 5? Eplain.. WRITING Eplain wh the epression 0 4 is undefined. EXAMPLE on p. 503 for Es. 3 4 EVALUATING EXPRESSIONS Evaluate the epression (3) 6. () (4) EXAMPLE on p. 504 for Es p p ( ) 5 8. (3 ) p p ERROR ANALYSIS Describe and correct the error in evaluating the epression 6 p p p EXAMPLE 3 on p. 505 for Es SIMPLIFYING EXPRESSIONS Simplif the epression. Write our answer using onl positive eponents (4g) 3 3. (h) m 3 n (6 3 ) (5fg ) r s (z) 4. 9 (3d) 3 4. (3) MULTIPLE CHOICE Which epression simplifies to 4? A 4 B 3 () 4 C 4 D MULTIPLE CHOICE Which epression is equivalent to (4 p 0 p 3)? A B 44 C 0 D Chapter 8 Eponents and Eponential Functions

22 CHALLENGE In Eercises 4648, tell whether the statement is true for all nonzero values of a and b. If it is not true, give a countereample. 46. a 3 a 4 5 a 47. a b 5 b a 48. a b 5 a b 49. REASONING For n > 0, what happens to the value of a n as n increases? PROBLEM SOLVING EXAMPLE 4 on p. 505 for Es MASS The mass of a grain of salt is about 0 4 gram. About how man grains of salt are in a bo containing 00 grams of salt? 5. MASS The mass of a grain of a certain tpe of rice is about 0 gram. About how man grains of rice are in a bo containing 0 3 grams of rice? 5. BOTANY The average mass of the fruit of the wolffia angusta plant is about 0 4 gram. The largest pumpkin ever recorded had a mass of about 0 4 kilograms. About how man times greater is the mass of the largest pumpkin than the mass of the fruit of the wolffia angusta plant? 53. MEDICINE A doctor collected about 0 liter of blood from a patient to run some tests. The doctor determined that a drop of the patient s blood, or about 0 6 liter, contained about 0 7 red blood cells. How man red blood cells did the entire sample contain? 54. SHORT RESPONSE One of the smallest plant seeds comes from an orchid, and one of the largest plant seeds comes from a giant fan palm. A seed from an orchid has a mass of 0 9 gram and is 0 3 times less massive than a seed from a giant fan palm. A student sas that the seed from the giant fan palm has a mass of about kilogram. Is the student correct? Eplain. Orchid Giant fan palm 55. MULTIPLE REPRESENTATIONS Consider folding a piece of paper in half a number of times. a. Making a Table Each time the paper is folded, record the number of folds and the fraction of the original area in a table like the one shown. Number of folds 0 3 Fraction of original area???? b. Writing an Epression Write an eponential epression for the fraction of the original area of the paper using a base of. 8.3 Define and Use Zero and Negative Eponents 507

23 56. SCIENCE Diffusion is the movement of molecules from one location to another. The time t (in seconds) it takes molecules to diffuse a distance of centimeters is given b t 5 where D is the diffusion coefficient. D a. You can eamine a cross section of a drop of ink in water to see how the ink diffuses. The diffusion coefficient for the molecules in the drop of ink is about 0 5 square centimeter per second. How long will it take the ink to diffuse micrometer (0 4 centimeter)? b. Check our answer to part (a) using unit analsis. 57. EXTENDED RESPONSE The intensit of sound I (in watts per square meter) can be modeled b I Pd where P is the power (in watts) of the sound s source and d is the distance (in meters) that ou are from the source of the sound. Not to scale a. What is the power (in watts) of the siren of the firetruck shown in the diagram? b. Using the power of the siren ou found in part (a), simplif the formula for the intensit of sound from the siren. c. Eplain what happens to the intensit of the siren when ou double our distance from it. 58. CHALLENGE Coal can be burned to generate energ. The heat energ in pound of coal is about 0 4 BTU (British Thermal Units). Suppose ou have a stereo. It takes about 0 pounds of coal to create the energ needed to power the stereo for ear. a. About how man BTUs does our stereo use in ear? b. Suppose the power plant that delivers energ to our home produces 0 pound of sulfur dioide for each 0 6 BTU of energ that it creates. How much sulfur dioide is added to the air b generating the energ needed to power our stereo for ear? MIXED REVIEW PREVIEW Prepare for Lesson 8.4 in Es Evaluate the epression p 0 3 (p. 489) p 0 5 (p. 489) (p. 495) (p. 495) 0 3 Solve the linear sstem. Then check our answer. (pp. 47, 435, 444, 45) EXTRA PRACTICE for Lesson 8.3, p. 945 ONLINE QUIZ at classzone.com

24 Etension Use after Lesson 8.3 Ke Vocabular cube root Define and Use Fractional Eponents GOAL Use fractional eponents. In Lesson.7, ou learned to write the square root of a number using a radical sign. You can also write a square root of a number using eponents. For an a 0, suppose ou want to write Ï a as a k. Recall that a number b (in this case, a k ) is a square root of a number a provided b 5 a. Use this definition to find a value for k as follows. b 5 a Definition of square root (a k ) 5 a Substitute a k for b. a k 5 a Product of powers propert Because the bases are the same in the equation a k 5 a, the eponents must be equal: k 5 Set eponents equal. k 5 Solve for k. So, for a nonnegative number a, Ï a 5 a. You can work with eponents of and multiples of just as ou work with integer eponents. E XAMPLE Evaluate epressions involving square roots a. 6 5 Ï 6 b Ï c () p 5 d. 4 3 () p (3) Ï Ï FRACTIONAL EXPONENTS You can work with other fractional eponents just as ou did with. Etension: Define and Use Fractional Eponents 509

25 CUBE ROOTS If b 3 5 a, then b is the cube root of a. For eample, 3 5 8, so is the cube root of 8. The cube root of a can be written as 3 Ï a or a 3. E XAMPLE Evaluate epressions involving cube roots a Ï 3 7 b Ï Ï c (3) p 4 d. 5 3 (3) p () Ï Ï PROPERTIES OF EXPONENTS The properties of eponents for integer eponents also appl to fractional eponents. E XAMPLE 3 Use properties of eponents a. p 5 5 () (5/) b. 643 p (43) / (73) (3) PRACTICE EXAMPLES,, and 3 on pp for Es. Evaluate the epression p p REASONING Show that the cube root of a can be written as a 3 using an argument similar to the one given for square roots on the previous page. 50 Chapter 8 Eponents and Eponential Functions

26 MIXED REVIEW of Problem Solving Lessons GRIDDED ANSWER In 004 the fastest computers could record about 0 9 bits per second. (A bit is the smallest unit of memor storage for computers.) Scientists believed that the speed limit at the time was about 0 bits per second. About how man times more bits per second was the speed limit than the fastest computers? STATE TEST PRACTICE classzone.com 4. GRIDDED ANSWER The least intense sound that is audible to the human ear has an intensit of about 0 watt per square meter. The intensit of sound from a jet engine at a distance of 30 meters is about 0 5 times greater than the least intense sound. Find the intensit of sound from the jet engine.. MULTI-STEP PROBLEM An office suppl store sells cubical containers that can be used to store paper clips, rubber bands, or other supplies. a. One of the containers has a side length of 4 inches. Find the container s volume b writing the side length as an improper fraction and substituting the length into the formula for the volume of a cube. b. Identif the propert of eponents ou used to find the volume in part (a). 3. SHORT RESPONSE Clouds contain millions of tin spherical water droplets. The radius of one droplet is shown. a. Find the order of magnitude of the volume of the droplet. b. Droplets combine to form raindrops. The radius of a raindrop is about 0 times greater than the droplet s radius. Find the order of magnitude of the volume of the raindrop. c. Eplain how ou can find the number of droplets that combine to form the raindrop. Then find the number of droplets and identif an properties of eponents ou used. r r = 0 4 cm 5. EXTENDED RESPONSE For an eperiment, a scientist dropped a spoonful, or about 0 cubic inch, of biodegradable olive oil into a pond to see how the oil would spread out over the surface of the pond. The scientist found that the oil spread until it covered an area of about 0 5 square inches. a. About how thick was the laer of oil that spread out across the pond? Check our answer using unit analsis. b. The pond has a surface area of 0 7 square inches. If the oil spreads to the same thickness as in part (a), how man cubic inches of olive oil would be needed to cover the entire surface of the pond? c. Eplain how ou could find the amount of oil needed to cover a pond with a surface area of 0 square inches. 6. OPEN-ENDED The table shows units of measurement of time and the durations of the units in seconds. Name of unit Duration (seconds) Gigasecond 0 9 Megasecond 0 6 Millisecond 0 3 Nanosecond 0 9 a. Use the table to write a conversion problem that can be solved b appling a propert of eponents involving products. b. Use the table to write a conversion problem that can be solved b appling a propert of eponents involving quotients. Mied Review of Problem Solving 5

27 8.4 Use Scientific Notation Before You used properties of eponents. Now You will read and write numbers in scientific notation. Wh? So ou can compare lengths of insects, as in E. 5. Ke Vocabular scientific notation Numbers such as,000,000, 53,000, and are written in standard form. Another wa to write a number is to use scientific notation. KEY CONCEPT For Your Notebook Scientific Notation A number is written in scientific notation when it is of the form c 3 0 n where c < 0 and n is an integer. Number Standard form Scientific notation Two million Five thousandths,000, E XAMPLE Write numbers in scientific notation a. 4,590, Move decimal point 7 places to the left. Eponent is 7. b Move decimal point 5 places to the right. Eponent is 5. E XAMPLE Write numbers in standard form READING A positive number in scientific notation is greater than if the eponent is positive. A positive number in scientific notation is between 0 and if the eponent is negative. a ,007,500 Eponent is 6. Move decimal point 6 places to the right. b Eponent is 4. Move decimal point 4 places to the left. at classzone.com GUIDED PRACTICE for Eamples and. Write the number 539,000 in scientific notation. Then write the number in standard form. 5 Chapter 8 Eponents and Eponential Functions

28 E XAMPLE 3 Order numbers in scientific notation Order 03,400,000, , and 80,760,000 from least to greatest. Solution STEP Write each number in scientific notation, if necessar. 03,400, ,760, STEP Order the numbers. First order the numbers with different powers of 0. Then order the numbers with the same power of 0. Because 0 7 < 0 8, ou know that is less than both and Because.034 < 7.8, ou know that is less than So, < < STEP 3 Write the original numbers in order from least to greatest. 80,760,000; 03,400,000; E XAMPLE 4 Compute with numbers in scientific notation Evaluate the epression. Write our answer in scientific notation. a. ( )( ) AVOID ERRORS Notice that is not written in scientific notation because 4.45 > 0. 5 (8.5 p.7) 3 (0 p 0 6 ) Commutative propert and associative propert Product of powers propert 5 ( ) Write 4.45 in scientific notation ( ) Associative propert Product of powers propert b. ( ) (0 3 ) Power of a product propert Power of a power propert REVIEW FRACTIONS For help with fractions, see p. 95. c Product rule for fractions Quotient of powers propert 5 ( ) Write 0.75 in scientific notation ( ) Associative propert Product of powers propert GUIDED PRACTICE for Eamples 3 and 4. Order.7 0 5, , and 7,500 from least to greatest. Evaluate the epression. Write our answer in scientific notation. 3. ( ) ( )( ) Use Scientific Notation 53

29 E XAMPLE 5 Solve a multi-step problem BLOOD VESSELS Blood flow is partiall controlled b the cross-sectional area of the blood vessel through which the blood is traveling. Three tpes of blood vessels are venules, capillaries, and arterioles. Capillar Venule Arteriole r r r r = mm r =.0 0 mm r = mm a. Let r be the radius of a venule, and let r be the radius of a capillar. Find the ratio of r to r. What does the ratio tell ou? b. Let A be the cross-sectional area of a venule, and let A be the cross-sectional area of a capillar. Find the ratio of A to A. What does the ratio tell ou? c. What is the relationship between the ratio of the radii of the blood vessels and the ratio of their cross-sectional areas? Solution a. From the diagram, ou can see that the radius of the venule r is millimeter and the radius of the capillar r is millimeter. r r ANOTHER WAY You can also find the ratio of the crosssectional areas b finding the areas using the values for r and r, setting up a ratio, and then simplifing. The ratio tells ou that the radius of the venule is twice the radius of the capillar. b. To find the cross-sectional areas, use the formula for the area of a circle. A 5 πr A πr 5 r r Write ratio. Divide numerator and denominator b p. 5 r r Power of a quotient propert Substitute and simplif. The ratio tells ou that the cross-sectional area of the venule is four times the cross-sectional area of the capillar. c. The ratio of the cross-sectional areas of the blood vessels is the square of the ratio of the radii of the blood vessels. GUIDED PRACTICE for Eample 5 6. WHAT IF? Compare the radius and cross-sectional area of an arteriole with the radius and cross-sectional area of a capillar. 54 Chapter 8 Eponents and Eponential Functions

30 8.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3, 7, and 53 5 STANDARDIZED TEST PRACTICE Es., 5, 48, 49, 54, and 59 5 MULTIPLE REPRESENTATIONS E. 58. VOCABULARY Is written in scientific notation? Eplain wh or wh not.. WRITING Is between 0 and or greater than? Eplain how ou know. EXAMPLE on p. 5 for Es. 3 5 WRITING IN SCIENTIFIC NOTATION Write the number in scientific notation ,000, ,065, ,060,000, ,000,000,000, MULTIPLE CHOICE Which number represents 54,004,000,000 written in scientific notation? A B C D EXAMPLE on p. 5 for Es. 6 8 WRITING IN STANDARD FORM Write the number in standard form ERROR ANALYSIS Describe and correct the error in writing in standard form EXAMPLE 3 on p. 53 for Es. 9 3 ORDERING NUMBERS Order the numbers from least to greatest ,000; ;,439; ,000,000; ; ; 55,004,000; ; ; ; ; ; ; ; ; ; COMPARING NUMBERS Cop and complete the statement using <, >, or ? 56, ,000.? ? ? ? ,589,000? Use Scientific Notation 55

31 EXAMPLE 4 on p. 53 for Es EVALUATING EXPRESSIONS Evaluate the epression. Write our answer in scientific notation. 39. ( )( ) 40. ( )( ) 4. ( )( ) ( ) ( ) ( ) 48. MULTIPLE CHOICE Which number is the value of ? A B C D OPEN ENDED Write two numbers in scientific notation whose product is Write two numbers in scientific notation whose quotient is CHALLENGE Add the numbers and without writing the numbers in standard form. Write our answer in scientific notation. Describe the steps ou take. PROBLEM SOLVING EXAMPLE 3 on p. 53 for Es INSECT LENGTHS The lengths of several insects are shown in the table. a. List the lengths of the insects in order from least to greatest. b. Which insects are longer than the fringed ant beetle? Insect Length (millimeters) Fringed ant beetle Walking stick 555 Parasitic wasp Elephant beetle ASTRONOMY The spacecrafts Voager and Voager were launched in 977 to gather data about our solar sstem. As of March, 004, Voager had traveled a total distance of about 9,643,000,000 miles, and Voager had traveled a total distance of about miles. Which spacecraft had traveled the greater distance at that time? EXAMPLE 4 on p. 53 for E. 53 EXAMPLE 5 on p. 54 for Es AGRICULTURE In 00, about pounds of cotton were produced in California. The cotton was planted on acres of land. What was the average number of pounds of cotton produced per acre? Round our answer to the nearest whole number. 54. SHORT RESPONSE The average flow rate of the Amazon River is about cubic feet per second. The average flow rate of the Mississippi River is about cubic feet per second. Find the ratio of the flow rate of the Amazon to the flow rate of the Mississippi. Round to the nearest whole number. What does the ratio tell ou? 56 5 WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

32 55. ASTRONOMY The radius of Earth and the radius of the moon are shown. a. Find the ratio of the radius of Earth to the radius of the moon. Round to the nearest hundredth. What does the ratio tell ou? b. Assume Earth and the moon are spheres. Find the ratio of the volume of Earth to the volume of the moon. Round to the nearest hundredth. What does the ratio tell ou? c. What is the relationship between the ratios of the radii and the ratios of the volumes? 56. MULTI-STEP PROBLEM In 954, 50 swarms of locusts were observed in Kena. The largest swarm covered an area of 00 square kilometers. The average number of locusts in a swarm is about locusts per square kilometer. a. About how man locusts were in Kena s largest swarm? Write our answer in scientific notation. b. The average mass of a desert locust is grams. What was the total mass (in kilograms) of Kena s largest swarm? Write our answer in scientific notation. 57. DIGITAL PHOTOGRAPHY When a picture is taken with a digital camera, the resulting image is made up of square piels (the smallest unit that can be displaed on a monitor). For one image, the side length of a piel is inch. A print of the image measures piels b piels. What are the dimensions of the print in inches? 58. MULTIPLE REPRESENTATIONS The speed of light is miles per second. a. Writing an Epression Assume ear is 365 das. Write an epression to convert the speed of light from miles per second to miles per ear. b. Making a Table Make a table that shows the distance light travels in, 0, 00, 000, 0,000, and 00,000 ears. Our gala has a diameter of about miles. Based on the table, about how long would it take for light to travel across our gala? 59. EXTENDED RESPONSE When a person is at rest, approimatel liter of blood flows through the heart with each heartbeat. The human heart beats about 70 times per minute. a. Calculate About how man liters of blood flow through the heart each minute when a person is at rest? b. Estimate There are approimatel minutes in a ear. Use our answer from part (a) to estimate the number of liters of blood that flow through the human heart in ear, in 0 ears, and in 80 ears. Write our answers in scientific notation. c. Eplain Are our answers to part (b) underestimates oroverestimates? Eplain. 8.4 Use Scientific Notation 57

33 60. CHALLENGE A solar flare is a sudden eruption of energ in the sun s atmosphere. Solar flares are classified according to their peak X-ra intensit (in watts per meter squared) and are denoted with a capital letter and a number, as shown in the table. For eample, a C4 flare has a peak intensit of watt per square meter. Class Bn Cn Mn Xn Peak intensit (w/m ) n n n n a. In November 003, a massive X45 solar flare was observed. In April 004, a C9 flare was observed. How man times greater was the intensit of the X45 flare than that of the C9 flare? b. A solar flare ma be accompanied b a coronal mass ejection (CME), a bubble of mass ejected from the sun. A CME related to the X45 flare was estimated to be traveling at 8. million kilometers per hour. At that rate, how long would it take the CME to travel from the sun to Earth, a distance of about meters? MIXED REVIEW PREVIEW Prepare for Lesson 8.5 in Es Write the percent as a decimal. (p. 96) 6. 33% % % % % % % % Graph the equation (p. 5) (p. 5) (p. 5) 7. 5 (p. 5) (p. 44) (p. 44) QUIZ for Lessons Simplif the epression. Write our answer using onl positive eponents. (p. 503). (4) 4 p (4) 6. (3 7 ) 3 3. (5z) 3 4. (6) Write the number in standard form. (p. 5) DINOSAURS The estimated masses of several dinosaurs are shown in the table. (p. 5) a. List the masses of the dinosaurs in order from least to greatest. b. Which dinosaurs are more massive than Brachiosaurus? Dinosaur Mass (kilograms) Brachiosaurus 77,00 Diplodocus Apatosaurus 9,900 Ultrasaurus EXTRA PRACTICE for Lesson 8.4, p. 945 ONLINE QUIZ at classzone.com

34 Graphing Calculator ACTIVITY Use after Lesson Use Scientific Notation classzone.com Kestrokes QUESTION How can ou use a graphing calculator to solve problems that involve numbers in scientific notation? E XAMPLE Use numbers in scientific notation Gold is one of man trace elements dissolved in seawater. There is about gram of gold per kilogram of seawater. The mass of the oceans is about kilograms. About how much gold is present in the oceans? STEP Write a verbal model Amount of gold present in oceans (grams) 5 Amount of gold in kilogram of seawater (gram/kilogram) p Amount of seawater in oceans (kilograms) STEP Find product The product is ( ) p ( ) STEP 3 Read result The calculator indicates that a number is in scientific notation b using E. You can read the calculator s result.54e3 as There are about grams of gold present in the oceans. (.*0^-8)(.4*0 ^).54E3 P RACTICE Evaluate the epression. Write the result in scientific notation.. ( )( ). ( )( ) 3. ( ) 4 ( ) 4. ( ) 4 ( ) 5. GASOLINE A scientist estimates that it takes about grams of carbon from ancient plant matter to produce gallon of gasoline. In 00 motor vehicles in the U.S. used about gallons of gasoline. a. If all of the gasoline used in 00 b motor vehicles in the U.S. came from carbon from ancient plant matter, how man grams of carbon were used to produce the gasoline? b. There are about atoms of carbon in gram of carbon. How man atoms of carbon were used? 8.4 Use Scientific Notation 59

35 8.5 Write and Graph Eponential Growth Functions Before Now You wrote and graphed linear models. You will write and graph eponential growth models. Wh? So ou can find the value of a collector car, as in Eample 4. Ke Vocabular eponential function eponential growth compound interest An eponential function is a function of the form 5 ab where a Þ 0, b > 0, and b Þ. Eponential functions are nonlinear functions. Observe how an eponential function compares with a linear function. Linear function: 5 3 Eponential function: 5 p E XAMPLE Write a function rule Write a rule for the function Solution STEP Tell whether the function is eponential Here, the -values are multiplied b for each increase of in, so the table represents an eponential function of the form 5 ab where b 5. STEP Find the value of a b finding the value of when 5 0. When 5 0, 5 ab 0 5 a p 5 a. The value of when 5 0 is 8, so a 5 8. STEP 3 Write the function rule. A rule for the function is 5 8 p. GUIDED PRACTICE for Eample. Write a rule for the function Chapter 8 Eponents and Eponential Functions

36 E XAMPLE Graph an eponential function READ A GRAPH Notice that the graph has a -intercept of and that it gets closer to the negative -ais as the -values decrease. Graph the function 5. Identif its domain and range. Solution STEP Make a table b choosing a few values for and finding the values of. The domain is all real numbers. 5 0 s, d (, ) 4 s, 4 4 d (0, ) STEP Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, ou can see that the range is all positive real numbers. 4 (, 4) E XAMPLE 3 Compare graphs of eponential functions Graph the functions 5 3 p and 53 p. Compare each graph with the graph of 5. Solution To graph each function, make a table of values, plot the points, and draw a smooth curve through the points p 53 p ? 5 3? Because the -values for 5 3 p are 3 times the corresponding -values for 5, the graph of 5 3 p is a vertical stretch of the graph of 5. Because the -values for 53 p are 3 times the corresponding -values for 5, the graph of 53 p is a vertical stretch with a reflection in the -ais of the graph of GUIDED PRACTICE for Eamples and 3. Graph 5 5 and identif its domain and range. 3. Graph 5 3 p. Compare the graph with the graph of Graph 5 3 p. Compare the graph with the graph of Write and Graph Eponential Growth Functions 5

37 EXPONENTIAL GROWTH When a > 0 and b >, the function 5 ab represents eponential growth. When a quantit grows eponentiall, it increases b the same percent over equal time periods. To find the amount to which the quantit grows after t time periods, use the following model. KEY CONCEPT For Your Notebook REWRITE EQUATIONS Notice that ou can rewrite 5 ab as 5 a( r) t b replacing b with r and with t (for time). Eponential Growth Model a is the initial amount. r is the growth rate. 5 a( r) t r is the growth factor. t is the time period. Notice the relationship between the growth rate r and the growth factor r. If the initial amount of a quantit is a units and the quantit is growing at a rate of r, then after one time period the new amount is: Initial amount amount of increase 5 a r p a 5 a( r) ANOTHER WAY For alternative methods for solving Eample 4, turn to page 58 for the Problem Solving Workshop. E XAMPLE 4 Solve a multi-step problem COLLECTOR CAR The owner of a 953 Hudson Hornet convertible sold the car at an auction. The owner bought it in 984 when its value was $,000. The value of the car increased at a rate of 6.9% per ear. a. Write a function that models the value of the car over time. b. The auction took place in 004. What was the approimate value of the car at the time of the auction? Round our answer to the nearest dollar. Solution a. Let C be the value of the car (in dollars), and let t be the time (in ears) since 984. The initial value a is $,000, and the growth rate r is C 5 a( r) t Write eponential growth model. AVOID ERRORS The growth rate in this eample is 6.9%, or So, the growth factor is 0.069, or.069, not ,000( 0.069) t Substitute,000 for a and for r. 5,000(.069) t Simplif. b. To find the value of the car in 004, 0 ears after 984, substitute 0 for t. C 5,000(.069) 0 Substitute 0 for t. ø 4,778 Use a calculator. c In 004 the value of the car was about $4,778. at classzone.com 5 Chapter 8 Eponents and Eponential Functions

38 COMPOUND INTEREST Compound interest is interest earned on both an initial investment and on previousl earned interest. Compounding of interest can be modeled b eponential growth where a is the initial investment, r is the annual interest rate, and t is the number of ears the mone is invested. E XAMPLE 5 Standardized Test Practice You put $50 in a savings account that earns 4% annual interest compounded earl. You do not make an deposits or withdrawals. How much will our investment be worth in 5 ears? A $300 B $304.6 C $ D $78,50 ESTIMATE You can use the simple interest formula, I 5 prt, to estimate the amount of interest earned: (50)(0.04)(5) Compounding interest will result in slightl more than $50. Solution 5 a( r) t Write eponential growth model. 5 50( 0.04) 5 Substitute 50 for a, 0.04 for r, and 5 for t. 5 50(.04) 5 Simplif. ø Use a calculator. You will have $304.6 in 5 ears. c The correct answer is B. A B C D GUIDED PRACTICE for Eamples 4 and 5 5. WHAT IF? In Eample 4, suppose the owner of the car sold it in 994. Find the value of the car to the nearest dollar. 6. WHAT IF? In Eample 5, suppose the annual interest rate is 3.5%. How much will our investment be worth in 5 ears? 8.5 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3 and 4 5 STANDARDIZED TEST PRACTICE Es. 3, 8, 34, 35, 4, 43, 46, and 50 5 MULTIPLE REPRESENTATIONS E. 4 SKILL PRACTICE. VOCABULARY In the eponential growth model 5 a( r) t, the quantit r is called the?.. VOCABULARY For what values of b does the eponential function 5 ab (where a > 0) represent eponential growth? 3. WRITING How does the graph of 5 p 5 compare with the graph of 5 5? Eplain. 8.5 Write and Graph Eponential Growth Functions 53

39 EXAMPLE on p. 50 for Es. 4 8 WRITING FUNCTIONS Write a rule for the function WRITING Given a table of values, describe how can ou tell if the table represents a linear function or an eponential function. EXAMPLE on p. 5 for Es. 9 GRAPHING FUNCTIONS Graph the function and identif its domain and range (.5) 4. 5 (.5) 5. 5 (.) 6. 5 (4.3) ERROR ANALYSIS The price P (in dollars) of a pound of flour was $.7 in 999. The price has increased b about % each ear. Let t be the number of ears since 999. Describe and correct the error in finding the price of a pound of flour in 00. P 5 a( r) t 50.7( ) (3) In 00 the price of a pound of flour was $7.9. EXAMPLE 3 on p. 5 for Es. 34 COMPARING GRAPHS OF FUNCTIONS Graph the function. Compare the graph with the graph of p p p p p p p p p p p p MULTIPLE CHOICE The graph of which function is shown? A f() 5 6 C f() 5 3 p 6 B f() 5 3 D f() 5 6 p 3 3 s0, 3d (, ) 35. WRITING If a population triples each ear, what is the population s growth rate (as a percent)? Eplain. 36. CHALLENGE Write a linear function and an eponential function whose graphs pass through the points (0, ) and (, 6). 37. CHALLENGE Compare the graph of the function f() 5 with the graph of the function g() 5 4 p. Use properties of eponents to eplain our observations WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

40 PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLES 4 and 5 on pp for Es INVESTMENTS You deposit $5 in a savings account that earns 5% annual interest compounded earl. Find the balance in the account after the given amounts of time. a. ear b. ears c. 5 ears d. 0 ears 39. MULTI-STEP PROBLEM One computer industr epert reported that there were about 600 million computers in use worldwide in 00 and that the number was increasing at an annual rate of about 0%. a. Write a function that models the number of computers in use over time. b. Use the function to predict the number of computers that will be in use worldwide in MULTI-STEP PROBLEM A research association reported that 3,73,000 gas grills were shipped b various manufacturers in the U.S. in 985. Shipments increased b about 7% per ear from 985 to 00. a. Write a function that models the number of gas grills shipped over time. b. About how man gas grills were shipped in 00? 4. MULTIPLE REPRESENTATIONS A tree s cross-sectional area taken at a height of 4.5 feet from the ground is called its basal area and is measured in square inches. Tree growth can be measured b the growth of the tree s basal area. The initial basal area and annual growth rate for two particular trees are shown. a. Writing a Model Write a function that models the basal area A of each tree over time. b. Graphing a Function Use a graphing calculator to graph the functions from part (a) in the same coordinate plane. In about how man ears will the trees be the same height? 8.5 Write and Graph Eponential Growth Functions 55

41 4. SHORT RESPONSE A compan sells advertising blimps. The table shows the costs of advertising blimps of different lengths. Does the table represent an eponential function? Eplain. Length, l (feet) Cost, c (dollars) MULTIPLE CHOICE A weblog, or blog, refers to a website that contains a personal journal. According to one analst, over one 8 month period, the number of blogs in eistence doubled about ever 6 months. The analst estimated that there were about 600,000 blogs at the beginning of the period. How man blogs were there at the end of the period? A 660,000 B,00,000 C 4,800,000 D 6,00, TELECOMMUNICATIONS For the period 99 00, the number (in millions) of Internet users worldwide can be modeled b the function (.65) where is the number of ears since 99. a. Identif the initial amount, the growth factor, and the growth rate. b. Graph the function. Identif its domain and range. c. Use the graph to estimate the ear in which the number of Internet users worldwide was about million. 45. GRAPHING CALCULATOR The frequenc (in hertz) of a note plaed on a piano is a function of the position of the ke that creates the note. The position of some piano kes and the frequencies of the notes created b the kes are shown below. Use the eponential regression feature on a graphing calculator to find an eponential model for the frequenc of piano notes. What is the frequenc of the note created b the 30 th ke? 46. EXTENDED RESPONSE In 830, the population of the United States was,866,00. B 890, the population was 6,947,74. a. Model Assume the population growth from 830 to 890 was linear. Write a linear model for the U.S. population from 830 to 890. B about how much did the population grow per ear from 830 to 890? b. Model Assume the population growth from 830 to 890 was eponential. Write an eponential model for the U.S. population from 830 to 890. B approimatel what percent did the population grow per ear from 830 to 890? c. Eplain The U.S. population was 3,9,876 in 850 and 38,558,37 in 870. Which of the models in parts (a) and (b) is a better approimation of actual U.S. population for the time period ? Eplain STANDARDIZED TEST PRACTICE

42 COMPOUND INTEREST In Eercises 47 50, use the eample below to find the balance of the account compounded with the given frequenc. E XAMPLE Use the general compound interest formula FINANCE You deposit $000 in an account that pas 3% annual interest. Find the balance after 8 ears if the interest is compounded monthl. Solution The general formula for compound interest is A 5 P r n nt. In this formula, P is the initial amount, called principal, in an account that pas interest at an annual rate r and that is compounded n times per ear. The amount A (in dollars) is the amount in the account after t ears. Here, the interest is compounded monthl. So, n 5. A5 P r n nt Write compound interest formula (8) Substitute 000 for P, 0.03 for r, for n, and 8 for t (.005) 96 Simplif. ø Use a calculator. c The account balance after 8 ears will be about $ Yearl 48. Quarterl 49. Dail (n 5 365) 50. WRITING Which compounding frequenc ields the highest balance in the account in the eample above: monthl, earl, quarterl, or dail? Eplain wh this is so. 5. CHALLENGE You invest $500 in an account that earns interest compounded monthl. Use a table or graph to find the least annual interest rate (to the nearest tenth of a percent) that the account would have to earn if ou want to have a balance of $600 in 4 ears. MIXED REVIEW PREVIEW Prepare for Lesson 8.6 in Es Evaluate the epression (p. 495) (p. 495) (p. 495) (p. 495) (p. 503) (p. 503) (p. 503) (p. 503) Write an equation of the line shown. (p. 83) (5, ) (, 3) (0, 3) (4, 0) EXTRA PRACTICE for Lesson 8.5, p. 945 ONLINE QUIZ at classzone.com 57

43 LESSON 8.5 Using ALTERNATIVE METHODS Another Wa to Solve Eample 4, page 5 MULTIPLE REPRESENTATIONS In Eample 4 on page 5, ou saw how to solve a problem about the value of a collector car over time b using an eponential model. You can also solve the problem b using a spreadsheet. P ROBLEM COLLECTOR CAR The owner of a 953 Hudson Hornet convertible sold the car at an auction. The owner bought it in 984 when its value was $,000. The value of the car increased at a rate of 6.9% per ear. a. Write a function that models the value of the car over time. b. The auction took place in 004. What was the approimate value of the car at the time of the auction? Round our answer to the nearest dollar. M ETHOD Using a Spreadsheet An alternative approach is to use a spreadsheet. a. The model for the value of the car over time is C 5,000(.069) t, as shown in Eample 4 on page 5. b. You can find the value of the car in 004 b creating a spreadsheet. STEP Create a table showing the ears since 984 and the value of the car. Enter the car s value in 984. To find the value in an ear after 984, multipl the car s value in the preceding ear b the growth factor, as shown in cell B3 below. FORMAT A SPREADSHEET Format the spreadsheet so that calculations are rounded to decimal places. 3 A Years since 984, t 0 B Value, C (dollars) 000 5B*.069 STEP Find the value of the car in 004 b using the fill down feature until ou get to the desired cell. 3 A Years since 984, t B Value, C (dollars) c From the spreadsheet, ou can see the value of the car was about $4,778 in Chapter 8 Eponents and Eponential Functions

44 P ROBLEM M ETHOD WHAT IF? Suppose the owner decided to sell the car when it was worth about $8,000. In what ear did the owner sell the car? Using a Spreadsheet To solve the equation algebraicall, ou need to substitute 8,000 for C and solve for t, but ou have not et learned how to solve this tpe of equation. An alternative to the algebraic approach is using a spreadsheet. STEP Use the same spreadsheet as on the previous page. STEP Find when the value of the car is about $8, A Years since 984, t B Value, C (dollars) The value of the car is about $8,000 when t 5 4. c The owner sold the car in 998. P RACTICE. TRANSPORTATION In 997 the average intercit bus fare for a particular state was $0. For the period , the bus fare increased at a rate of about % each ear. a. Write a function that models the intercit bus fare for the period b. Find the intercit bus fare in 998. Use two different methods to solve the problem. c. In what ear was the intercit bus fare $8.0? Eplain how ou found our answer.. ERROR ANALYSIS Describe and correct the error in writing the function for part (a) of Eercise. Let b be the bus fare (in dollars) and t be the number of ears since 997. b 5 0(0.) t 3. TECHNOLOGY A computer s Central Processing Unit (CPU) is made up of transistors. One manufacturer released a CPU in Ma 997 that had 7.5 million transistors. The number of transistors in the CPUs sold b the compan increased at a rate of 3.9% per month. a. Write a function that models the number T (in millions) of transistors in the compan s CPUs t months after Ma 997. b. Use a spreadsheet to find the number of transistors in a CPU released b the compan in November HOUSING The value of a home in 00 was $50,000. The value of the home increased at a rate of about 6.5% per ear. a. Write a function that models the value of the home over time. b. Use a spreadsheet to find the ear in which the value of the home was about $00,000. Using Alternative Methods 59

45 Investigating g Algebra ACTIVITY Use before Lesson Eponential Models MATERIALS arn scissors QUESTION How can ou model a situation using an eponential function? E XPLORE Collect data so that ou can write eponential models STEP Fold and cut Take about ard of arn and consider it to be unit long. Fold it in half and cut, as shown. You are left with two pieces of arn, each half the length of the original piece of arn. STEP Cop and complete Cop the table. Notice that the row for stage has the data from Step. For each successive stage, fold all the pieces of arn in half and cut. Then record the number of new pieces and the length of each new piece until the table is complete. Stage Number of pieces Length of each new piece?? 3?? 4?? 5?? DRAW CONCLUSIONS Use our observations to complete these eercises. Use the data in the first and second columns of the table. a. Do the data represent an eponential function? Eplain how ou know. b. Write a function that models the number of pieces of arn at stage. c. Use the function to find the number of pieces of arn at stage 0.. Use the data in the first and third columns of the table. a. Do the data represent an eponential function? Eplain how ou know. b. Write a function that models the length of each new piece of arn at stage. c. Use the function to find the length of each new piece of arn at stage Chapter 8 Eponents and Eponential Functions

46 8.6 Write and Graph Eponential Deca Functions Before You wrote and graphed eponential growth functions. Now You will write and graph eponential deca functions. Wh? So ou can use a graph to solve a sports problem, as in E. 50. Ke Vocabular eponential deca A table of values represents an eponential function 5 ab provided successive -values are multiplied b b each time the -values increase b. E XAMPLE Write a function rule Tell whether the table represents an eponential function. If so, write a rule for the function. a The value of when 5 0 is 3, so a 5 3. The -values are multiplied b 3 for each increase of in, so the table represents an eponential function of the form 5 ab with b 5 3. The table represents the eponential function 5 3 p 3. b The -values are multiplied b for 4 each increase of in, so the table represents an eponential function of the form 5 ab with b 5. 4 The value of when 5 0 is, so a 5. The table represents the eponential function 5 4. GUIDED PRACTICE for Eample. Tell whether the table represents an eponential function. If so, write a rule 0 for the function Write and Graph Eponential Deca Functions 53

47 E XAMPLE Graph an eponential function READ A GRAPH Notice that the graph has a -intercept of and that it gets closer to the positive -ais as the -values increase. Graph the function 5 and identif its domain and range. Solution STEP Make a table of values. The domain is all real numbers. 0 4 STEP Plot the points. 4 (, 4) STEP 3 Draw a smooth curve through the points. From either the table or the graph, ou can see the range is all positive real numbers. 3 (, ) (0, ) s, d 5 c s, 4 d E XAMPLE 3 Compare graphs of eponential functions Graph the functions 5 3 p and 5 3 p. Compare each graph with the graph of 5. Solution p 5 3 p c 5 3? c ? c Because the -values for 5 3 p are 3 times the corresponding -values for 5, the graph of 5 3 p is a vertical stretch of the graph of 5. Because the -values for 5 p 3 are times the corresponding 3 -values for 5, the graph of 5 p 3 is a vertical shrink with reflection in the -ais of the graph of 5. GUIDED PRACTICE for Eamples and 3. Graph 5 (0.4) and identif its domain and range. 3. Graph 5 5 p (0.4). Compare the graph with the graph of 5 (0.4). 53 Chapter 8 Eponents and Eponential Functions

48 COMPARE GRAPHS When a > 0 and 0 < b <, the function 5 ab represents eponential deca. The graph of an eponential deca function falls from left to right. In comparison, the graph of an eponential growth function 5 ab where a > 0 and b > rises from the left. E XAMPLE 4 Classif and write rules for functions Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function. a. b. (, ) (0, 0) (0, 8) (, 4) ANALYZE GRAPHS For the function 5 ab, where 5 0, the value of is 5 a 0 5 a. This means that the graph of 5 ab has a -intercept of a. Solution a. The graph represents eponential growth ( 5 ab where b > ). The -intercept is 0, so a 5 0. Find the value of b b using the point (, ) and a ab Write function. 5 0 p b Substitute.. 5 b Solve. A function rule is 5 0(.). b. The graph represents eponential deca ( 5 ab where 0 < b < ). The -intercept is 8, so a 5 8. Find the value of b b using the point (, 4) and a ab Write function p b Substitute b Solve. A function rule is 5 8(0.5). GUIDED PRACTICE for Eample 4 4. The graph of an eponential function passes through the points (0, 0) and (, 8). Graph the function. Tell whether the graph represents eponential growth or eponential deca. Write a rule for the function. CONCEPT SUMMARY For Your Notebook Eponential Growth and Deca Eponential Growth Eponential Deca 5 ab, a > 0 and b > 5 ab, a > 0 and 0 < b < (0, a) (0, a) 8.6 Write and Graph Eponential Deca Functions 533

49 EXPONENTIAL DECAY When a quantit decas eponentiall, it decreases b the same percent over equal time periods. To find the amount of the quantit left after t time periods, use the following model. KEY CONCEPT For Your Notebook REWRITE EQUATIONS Notice that ou can rewrite 5 ab as 5 a( r) t b replacing b with r and with t (for time). Eponential Deca Model a is the initial amount. r is the deca rate. 5 a( r) t 5 r is the deca factor. t is the time period. The relationship between the deca rate r and the deca factor r is similar to the relationship between the growth rate and growth factor in an eponential growth model. You will eplore this relationship in Eercise 45. E XAMPLE 5 Solve a multi-step problem FORESTRY The number of acres of Ponderosa pine forests decreased in the western United States from 963 to 00 b 0.5% annuall. In 963 there were about 4 million acres of Ponderosa pine forests. a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time. b. To the nearest tenth, about how man million acres of Ponderosa pine forests were there in 00? Solution AVOID ERRORS The deca rate in this eample is 0.5%, or So, the deca factor is 0.005, or 0.995, not a. Let P be the number of acres (in millions), and let t be the time (in ears) since 963. The initial value is 4, and the deca rate is P 5 a( r) t Write eponential deca model. 5 4( 0.005) t Substitute 4 for a and for r. 5 4(0.995) t Simplif. b. To find the number of acres in 00, 39 ears after 963, substitute 39 for t. P 5 4(0.995) 39 < 33.7 Substitute 39 for t. Use a calculator. c There were about 33.7 million acres of Ponderosa pine forests in 00. at classzone.com GUIDED PRACTICE for Eample 5 5. WHAT IF? In Eample 5, suppose the deca rate of the forests remains the same beond 00. About how man acres will be left in 00? 534 Chapter 8 Eponents and Eponential Functions

50 8.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7 and 49 5 STANDARDIZED TEST PRACTICE Es., 9, 36, 45, and 49 5 MULTIPLE REPRESENTATIONS E. 50. VOCABULARY What is the deca factor in the eponential deca model 5 a( r) t?. WRITING Eplain how ou can tell if a graph represents eponential growth or eponential deca. EXAMPLE on p. 53 for Es. 3 6 WRITING FUNCTIONS Tell whether the table represents an eponential function. If so, write a rule for the function EXAMPLE on p. 53 for Es. 7 8 GRAPHING FUNCTIONS Graph the function and identif its domain and range (0.3) 4. 5 (0.5) 5. 5 (0.) 6. 5 (0.9) 7. 5 (0.7) 8. 5 (0.5) EXAMPLE 3 on p. 53 for Es MULTIPLE CHOICE The graph of which function is shown? A 5 (0.5) B 5 (0.5) C p (0.5) D 5 4 p (0.5) (0, 4) (, ) COMPARING FUNCTIONS Graph the function. Compare the graph with the graph of p p 4. 5 p p p p p p p p p p Write and Graph Eponential Deca Functions 535

51 MATCHING Match the function with its graph (0.) (0.) (0.) A. B. C POPULATION A population of 90,000 decreases b.5% per ear. Identif the initial amount, the deca factor, and the deca rate. Then write a function that models the population over time. 36. MULTIPLE CHOICE What is the deca rate of the function 5 4(0.97) t? A 4 B 0.97 C 0.3 D ERROR ANALYSIS In 004 a person purchased a car for $5,000. The value of the car decreased b 4% annuall. Describe and correct the error in writing a function that models the value of the car since a( r) t 5 5,000(0.4) t EXAMPLE 4 on p. 533 for Es RECOGNIZING EXPONENTIAL MODELS Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function (0, 6) (, 4.8) (0, 8) (, 4.8) (0, 8) (,.8) at classzone.com 4. REASONING Without graphing, eplain how the graphs of the given functions are related to the graph of f() 5 (0.5). a. m() 5 3 p (0.5) b. n() 54 p (0.5) c. p() 5 (0.5) CHALLENGE Write an eponential function of the form 5 ab whose graph passes through the given points. 4. (0, ),, (, 0), (, 4) 44., 3,, WRITING The initial amount of a quantit is a units and the quantit is decaing at a rate of r (a percent per time period). Show that the amount of the quantit after one time period is a( r). Eplain how ou found our answer. 46. CHALLENGE Compare the graph of the function f() 5 4 with the graph of the function g() 5 p 4. Use properties of eponents to 6 eplain our observation WORKED-OUT SOLUTIONS for on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

52 PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLE 5 on p. 534 for Es CELL PHONES You purchase a cell phone for $5. The value of the cell phone decreases b about 0% annuall. Write a function that models the value of the cell phone over time. Then find the value of the cell phone after 3 ears. Round to the nearest dollar. 48. ANIMAL POPULATION Scientists studied the population of a species of bat in some caves in Missouri from 983 to 003. In 983, there were 4,00 bats living in the caves. That number decreased b about % annuall until 003. a. Identif the initial amount, the deca factor, and the deca rate. b. Write a function that models the number of bats since 983. Then find the number of bats in SHORT RESPONSE In 003 a famil bought a boat for $4000. The boat depreciates (loses value) at a rate of 7% annuall. In 006 a person offers to bu the boat for $3000. Should the famil sell the boat? Eplain. 50. MULTIPLE REPRESENTATIONS There are a total of 8 teams at the start of a citwide 3-on-3 basketball tournament. Half of the teams are eliminated after each round. a. Writing a Model Write a function for the number of teams left after rounds. b. Making a Table Make a table for the function using 5 0,,,..., 7. c. Drawing a Graph Use the table in part (b) to graph the function. After which round are there 4 teams left in the tournament? 5. GUITARS The frets on a guitar are the small metal bars that divide the fingerboard. The distance d (in inches) between the nut and the first fret or an two consecutive frets can be modeled b the function d 5.56(0.9439) f where f is the number of the fret farthest from the nut. Fret Nut a. Identif the deca factor and the deca rate for the model. b. What is the distance between the nut and the first fret? c. The distance between the th and 3th frets is about half the distance between the nut and the first fret. Use this fact to find the distance between the th and 3th frets. Use the model to verif our answer. 8.6 Write and Graph Eponential Deca Functions 537

53 5. CHALLENGE A college student finances a computer that costs $850. The financing plan states that as long as a minimum monthl pament of.5% of the remaining balance is made, the student does not have to pa interest for 4 months. The student makes onl the minimum monthl paments until the last pament. What is the amount of the last pament if the student bus the computer without paing interest? Round our answer to the nearest cent. 53. MULTI-STEP PROBLEM Maimal ogen consumption is the maimum volume of ogen (in liters per minute) that the bod uses during eercise. Maimal ogen consumption varies from person to person and decreases with age b about 0.5% per ear after age 5 for active adults. a. Model A 5-ear-old female athlete has a maimal ogen consumption of 4 liters per minute. Another 5-ear-old female athlete has a maimal ogen consumption of 3.5 liters per minute. Write a function for each athlete that models the maimal consumption each ear after age 5. b. Graph Graph the models in the same coordinate plane. c. Estimate About how old will the first athlete be when her maimal ogen consumption is equal to what the second athlete s maimal ogen consumption is at age 5? MIXED REVIEW PREVIEW Prepare for Lesson 9. in Es Simplif the epression. (p. 96) 54. (3) ( ) ( 4) ( 9) 60. (5 ) 6. (3 4)7 6. ( ) Solve the equation (p. 34) (p. 4) (p. 48) 66. 7( 5) 5 0 (p. 48) QUIZ for Lessons Graph the function (p. 50). 5 3 p 4 (p. 53) p 3 (p. 50) 4. 5 (0.) (p. 53) p 5 (p. 50) (0.4) (p. 53) 7. COINS You purchase a coin from a coin collector for $5. Each ear the value of the coin increases b 8%. Write a function that models the value of the coin over time. Then find the value of the coin after 0 ears. Round to the nearest cent. (p. 50) 538 EXTRA PRACTICE for Lesson 8.6, p. 945 ONLINE QUIZ at classzone.com

54 Etension Use after Lesson 8.6 Ke Vocabular geometric sequence common ratio Relate Geometric Sequences to Eponential Functions GOAL Identif, graph, and write geometric sequences. In a geometric sequence, the ratio of an term to the previous term is constant. This constant ratio is called the common ratio and is denoted b r. A geometric sequence with first term a and common ratio r has the form a, a r, a r, a r 3,.... For instance, if a 5 5 and r 5, the sequence 5, 5 p, 5 p, 5 p 3,..., or 5, 0, 0, 40,..., is geometric. E XAMPLE Identif a geometric sequence Tell whether the sequence is arithmetic or geometric. Then write the net term of the sequence. a. 3, 6, 9,, 5,... b. 8, 64, 3, 6, 8,... Solution a. The first term is a 5 3. Find the ratios of consecutive terms: a a a 3 a a 4 a a 5 a REVIEW ARITHMETIC SEQUENCES For help with identifing an arithmetic sequence and finding a common difference, see p Because the ratios are not constant, the sequence is not geometric. To see if the sequence is arithmetic, find the differences of consecutive terms. a a a 3 a a 4 a a 5 a The common difference is 3, so the sequence is arithmetic. The net term of the sequence is a 6 5 a b. The first term is a 5 8. Find the ratios of consecutive terms: a a a 3 a a 4 a a 5 a Because the ratios are constant, the sequence is geometric. The common ratio is. The net term of the sequence is a 6 5 a 5 p 5 4. E XAMPLE Graph a geometric sequence ANALYZE A GRAPH Notice that the graph in Eample appears to be eponential. To graph the sequence from part (b) of Eample, let each term s position number in the sequence be the -value. The term is the corresponding -value. Then make and plot the points Position, Term, Etension: Relate Geometric Sequences to Eponential Functions 539

55 FUNCTIONS The table shows that a rule for finding the nth term of a geometric sequence is a n 5 a r n. Notice that the rule is an eponential function. Position, n n Term, a n a a r a r a r 3... a r n For the nth term, ou multipl a b r (n ) times. p r p r p r p r KEY CONCEPT For Your Notebook General Rule for a Geometric Sequence The nth term of a geometric sequence with first term a and common ratio r is given b: a n 5 a r n. E XAMPLE 3 Write a rule for a geometric sequence Write a rule for the nth term of the geometric sequence in Eample. Then find a 0. Solution To write a rule for the nth term of the sequence, substitute the values for a and r in the general rule a n 5 a r n. Because a 5 8 and r 5, a n 5 8 p n. The 0th term of the sequence is a p PRACTICE EXAMPLES,, and 3 on pp for Es. 0 Tell whether the sequence is arithmetic or geometric. Then graph the sequence.. 3,, 48, 9,.... 7, 6, 5, 34, , 8,, 6, , 8, 6,, , 8, 36, 7, , 43, 57, 7,... Write a rule for the nth term of the geometric sequence. Then find a 7. 7., 5, 5, 5, , 6, 5, 04, , 7,,, A chain instructs the recipient to forward the to four more people. The table shows the number of rounds of sending the and the number of new s generated. Write a rule for the nth term of the sequence. Then graph the first si terms of the sequence. Number of rounds sending , n 3 4 Number of new s generated, a n Chapter 8 Eponents and Eponential Functions

56 MIXED REVIEW of Problem Solving Lessons MULTI-STEP PROBLEM The radius of the sun is about 96,600,000 kilometers. The radius of Earth is about 6370 kilometers. a. Write each radius in scientific notation. b. The surface area S of a sphere with radius r is given b S 5 4pr. Assume the sun and Earth are perfect spheres. Find their surface areas. Write our answers in scientific notation. c. What is the ratio of the surface area of the sun to the surface area of Earth? What does the ratio tell ou?. SHORT RESPONSE The graph shows the value of a truck over time. 30,000 (0, 5,000) 0,000 Value (dollars) 0,000 (, 3,750) Time (ears) a. Write an equation for the function whose graphisshown. b. At what rate is the truck losing value? Eplain. 3. GRIDDED ANSWER A new laptop computer costs $000. The value of the computer decreases over time. The value V (in dollars) of the computer after t ears is given b the function V 5 000(0.8) t. What is the deca rate, written as a decimal, of the value of the computer? 4. OPEN - ENDED The value of a house in Iowa increased, on average, at a rate of about 4% per quarter from the first quarter in 00 to the last quarter in 004. Write a function that models the value of the house over time. Choose an initial value of the house and a quarter such that the value of the house is about $75,000. STATE TEST PRACTICE classzone.com 5. EXTENDED RESPONSE A musician is saving mone to bu a new snare drum. The musician puts $00 in a savings account that pas 3% annual interest compounded earl. a. Write a function that models the amount of mone in the account over time. b. Graph the function. c. The musician wants a drum that costs $ Will there be enough in the account after 3 ears? Eplain. 6. MULTI-STEP PROBLEM The graph shows the value of a business over time. Value (dollars) 60,000 80,000 (0, 5,000) (, 9,500) Time (ears) a. Does the graph represent eponential growth or eponential deca? b. Write a function that models the value of the business over time. c. How much is the business worth after 4ears? 7. MULTI-STEP PROBLEM The half-life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient s bloodstream. A certain antibiotic has a half-life of about 8 hours. a. A patient is administered 500 milligrams of the medication. Write a function that models the amount of the medication in the patient s bloodstream over time. b. How much of the 500 milligram dose will be in the patient s bloodstream after 4 hours? Mied Review of Problem Solving 54

57 8 CHAPTER SUMMARY Big Idea BIG IDEAS For Your Notebook Appling Properties of Eponents to Simplif Epressions You can use the properties of eponents to simplif epressions. For the properties listed below, a and b are real numbers, and m and n are integers. Epression Propert a m p a n 5 a m n Product of powers propert (a m ) n 5 a mn Power of power propert (ab) 5 a m b m Power of product propert a m a n 5 a m n, a Þ Quotient of powers propert a b m 5 am b m, b Þ 0 Power of quotients propert Big Idea Working with Numbers in Scientific Notation You can write numbers in scientific notation. Number St andard form Scient ific notation Four billion 4, 000, 000, Thirt irt-two thousandths You can also compute with numbers in scientific notation. For eample: ( ) 3 ( ) , or 8,000,000 Big Idea 3 Writing and Graphing Eponential Functions You can write and graph eponential growth and deca functions. You can also model real-world situations involving eponential growth and eponential deca. Eponential growth Eponential deca Function: 5 ab, a > 0 and b > Function: 5 ab, a > 0 and 0 < b < Graph: Graph: (0, a) (0, a) Model: 5 a( r) t Model: 5 a( r) t 54 Chapter 8 Eponents and Eponential Functions

58 8 CHAPTER REVIEW REVIEW KEY VOCABULARY classzone.com Multi-Language Glossar Vocabular practice order of magnitude, p. 49 zero eponent, p. 503 negative eponent, p. 503 scientific notation, p. 5 eponential function, p. 50 eponential growth, p. 5 growth factor, growth rate, p. 5 compound interest, p. 53 eponential deca, p. 533 deca factor, deca rate, p. 534 VOCABULARY EXERCISES. Cop and complete: The function 5 00(0.3) t is an eponential? function, and the base 0.3 is called the?.. WRITING Eplain how ou can tell whether a table represents a linear function or an eponential function. Tell whether the function represents eponential growth or eponential deca. Eplain (0.85) 4. 5 (.0) 5. 5 (.) REVIEW EXAMPLES AND EXERCISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of Chapter Appl Eponent Properties Involving Products pp E XAMPLE Simplif (3 3 ) 4 p 5. (3 3 ) 4 p p ( 3 ) 4 p 5 Power of a product propert 5 8 p p 5 Power of a power propert Product of powers propert EXERCISES EXAMPLES,, 3, 4, and 5 on pp for Es. 6 5 Simplif the epression p (3) 7 (3) 8. z 3 p z 5 p z 5 9. ( 4 ) 5 0. [(7) 4 ] 4. [(b ) 8 ] 3. (6 4 p 3) 5 3. (8) 4. ( ) 4 p 5 5. EARTH SCIENCE The order of magnitude of the mass of Earth s atmosphere is 0 8 kilograms. The order of magnitude of the mass of Earth s oceans is 0 3 times greater. What is the order of magnitude of the mass of Earth s oceans? Chapter Review 543

59 8 CHAPTER REVIEW 8. Appl Eponent Properties Involving Quotients pp E XAMPLE Simplif 3 4 p p 5 (3 ) 4 p p 5 Power of a quotient propert Power of a power propert Multipl fractions Quotient of powers propert EXAMPLES,, and 3 on pp for Es. 6 4 EXERCISES Simplif the epression. 6. (3) 7 (3) p m n 3 9. p p p p r5 7r 0 s 5 4. PER CAPITA INCOME The order of magnitude of the population of Montana in 003 was 0 6 people. The order of magnitude of the total personal income (in dollars) for Montana in 003 was 0 0. What was the order of magnitude of the mean personal income in Montana in 003? 8.3 Define and Use Zero and Negative Eponents pp E XAMPLE Evaluate ( 0 5 ) 3. ( 0 5 ) p 0 p 5 Power of a power propert 5 8 p p 5 Definition of zero eponent Definition of negative eponents EXAMPLES,, and 4 on pp for Es. 5 9 EXERCISES Evaluate the epression p UNITS OF MEASURE Use the fact that femtogram kilogram and nanogram 5 0 kilogram to complete the following statement: nanogram 5? femtogram(s). 544 Chapter 8 Eponents and Eponential Functions

60 classzone.com Chapter Review Practice 8.4 Use Scientific Notation pp E XAMPLE Write the number in scientific notation. a Move decimal point left 3 places. Eponent is 3. b Move decimal point right 4 places. Eponent is 4. Write the number in standard form. a Eponent is. Move decimal point right places. b Eponent is 3. Move decimal point left 3 places. EXAMPLES,, 4, and 5 on pp for Es EXERCISES 30. Write 78,0 in scientific notation. 3. Write in standard form. Evaluate the epression. Write our answer in scientific notation. 3. ( )( ) MASS The mass m of a gate of the Thames Barrier in London is about kilograms. The mass m of the Great Pramid of Giza is about kilograms. Find the ratio of m to m. What does the ratio tell ou? 8.5 Write and Graph Eponential Growth Functions pp E XAMPLE Graph the function 5 4 and identif its domain and range. STEP Make a table. The domain is all real numbers. (, 6) 0 STEP Plot the points STEP 3 Draw a smooth curve through the points. s, 4 d 6 (, 4) (0, ) STEP 4 Identif the range. As ou can see from the graph, the range is all positive real numbers. EXAMPLES and 3 on p. 5 for Es EXERCISES Graph the function and identif its domain and range (.) (3.5) Graph the function 55 p. Compare the graph with the graph of 5. Chapter Review 545

61 8 CHAPTER REVIEW 8.6 Write and Graph Eponential Deca Functions pp E XAMPLE Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function. The graph represents eponential deca ( 5 ab where 0, b, ). The -intercept is, so a 5. Find the value of b b using the point (, 0.5) and a 5. 5 ab Write function. (0, ) (, 0.5) p b Substitute b Solve for b. A function rule is 5 (0.5). E XAMPLE CAR VALUE A famil purchases a car for $,000. The car depreciates (loses value) at a rate of about 6% annuall. Write a function that models the value of the car over time. Find the approimate value of the car in 4 ears. Let V represent the value (in dollars) of the car, and let t represent the time (in ears since the car was purchased). The initial value is,000, and the deca rate is 0.6. V5 a( r) t Write eponential deca model. 5,000( 0.6) t Substitute,000 for a and 0.6 for r. 5,000(0.84) t Simplif. To find the approimate value of the car in 4 ears, substitute 4 for t. V5,000(0.84) t 5,000(0.84) 4 ø $5477 The approimate value of the car in 4 ears is $5477. EXERCISES EXAMPLES 4 and 5 on pp for Es Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function (0, ) (, 4) (0, 3) (, ) 4. CAR VALUE The value of a car is $3,000. The car depreciates (loses value) at a rate of about 5% annuall. Write an eponential deca model for the value of the car. Find the approimate value of the car in 4 ears. 546 Chapter 8 Eponents and Eponential Functions

62 8 Simplif CHAPTER TEST the epression. Write our answer using eponents.. (6 p 7) 4. (3)(3) p (84 ) p 5 0 p [(4 3 )] 8. (5)0 (5) 3 Simplif the epression. 9. t p t 6 0. s t (6p) 3. (5) 4. p z 7 z9 5. ( 5 ) c Simplif the epression. Write our answer using onl positive eponents. 7. a3 3b d p (d)4 c p (8 6 3 ) 0. (5r 5 ) 3 p r Write the number in scientific notation ,94, Write the number in standard form Graph the function 5 4. Identif its domain and range. 38. Graph the function 5 p 4. Compare the graph with the graph of ANIMATION About btes of data make up a single frame of an animated film. There are 4 frames in second of a film. About how man btes of data are there in hour of an animated film? 40. SALARY A recent college graduate accepts a job at a law firm. The job has a salar of $3,000 per ear. The law firm guarantees an annual pa increase of 3% of the emploee s salar. a. Write a function that models the emploee s salar over time. Assume that the emploee receives onl the guaranteed pa increase. b. Use the function to find the emploee s salar after 5 ears. 4. SCIENCE At sea level, Earth s atmosphere eerts a pressure of atmosphere. Atmospheric pressure P (in atmospheres) decreases with altitude and can be modeled b P 5 ( ) a where a is the altitude (in meters). a. Identif the initial amount, deca factor, and deca rate. b. Use a graphing calculator to graph the function. c. Estimate the altitude at which the atmospheric pressure is about half of what it is at sea level. Chapter Test 547

63 8 Standardized TEST PREPARATION CONTEXT-BASED MULTIPLE CHOICE QUESTIONS Some of the information ou need to solve a contet-based multiple choice question ma appear in a table, a diagram, or a graph. P ROBLEM A scientist monitors bacteria cell growth in an eperiment. The scientist records the number of bacteria cells in a petri dish ever 0 minutes, as shown in the table. Number of 0 minute time periods, t Number of bacteria cells, c How man bacteria cells will there be after 3 hours? A B 7680 C 70 D 0 Plan INTERPRET THE TABLE Determine whether the table represents a linear or an eponential function. Use the information in the table to write a function. Then use the function to find the number of bacteria cells after 3 hours. STEP Determine whether the function is eponential. Solution t c STEP Write the function rule. STEP 3 Find the number of cells after 3 hours. Because the c-values are multiplied b for each increase of in t, the table represents an eponential function of the form c 5 ab t where b 5. The value of c when t 5 0 is 5, as shown in the table, so a 5 5. Substitute the values of a and b in the function c 5 ab t. A function rule is c 5 5 p t. There are 80 minutes in 3 hours. So there are nine 0 minute periods in 3 hours. Substitute 9 for t in the function rule ou wrote in Step. c 5 5 p t 5 5 p There are 7680 bacteria cells in the petri dish after 3 hours. The correct answer is B. ABCD 548 Chapter 8 Eponents and Eponential Functions

64 P ROBLEM A fish tank is a rectangular prism and is partiall filled with sand, as shown. The dimensions of the fish tank are given. The order of magnitude of the number of grains of sand in cubic inch is 0 3. Find the order of magnitude of the total number of grains of sand in the fish tank. 30 in. in. 5 in. 3 in. A 0 grains B 0 3 grains C 0 5 grains D 0 6 grains Plan INTERPRET THE DIAGRAM Use the information in the diagram to find the order of magnitude of the volume of sand in the fish tank. Multipl the volume b the order of magnitude of the number of grains of sand in cubic inch. STEP Find the order of magnitude of the volume of sand in the fish tank. Solution Use the formula for the volume of a rectangular prism. V 5 lwh Write formula for volume of rectangular prism p p 3 Substitute given values Multipl. STEP Find the order of magnitude of the number of grains of sand in the fish tank. The order of magnitude of the volume of sand is 0 3 cubic inches. Multipl the order of magnitude of the volume of sand in the fish tank b the order of magnitude of the grains of sand in cubic inch. 0 3 p The order of magnitude of the total number of grains of sand in the fish tank is 0 6. The correct answer is D. ABCD PRACTICE. In Problem, consider the section of the fish tank occupied b water onl. The order of magnitude of the weight of water per cubic inch is 0 pound. The tank is filled to the top. What is the order of magnitude of the weight of the water in the fish tank? A 0 8 pound B 0 6 pound C 0 pounds D 0 6 pounds. What is the volume of the clinder shown? 3 A 9π 3 B 3π 3 C 9π D 3π Standardized Test Preparation 549

65 8 Standardized TEST PRACTICE MULTIPLE CHOICE. The table represents which function? In Eercises 5 7, use the table below. Unit Number of meters Kilometer 0 3 A 5 3 p 5 C 5 3 p 5 B 5 3 p 5 D 53 p 5 Centimeter 0 Millimeter 0 3 Nanometer 0 9. What is the volume of the cube? A 4 3 B 3 C 6 D 64 3 In Eercises 3 and 4, use the table below. 3. List the elements in order from least concentration to greatest concentration. Element in Seawater Concentration (parts per million) Sulfur 904 Chloride Magnesium How man millimeters are in kilometer? A B 0 C 0 3 D How man nanometers are in a centimeter? A 0 B 0 7 C 0 7 D A micrometer is 0 3 times greater than a nanometer. How man meters are in a micrometer? A 0 7 B 0 C 0 6 D 0 6 In Eercises 8 and 9, use the graph below. 3 (0, ) (, 6) Sodium 0,770 A Sulfur, sodium, magnesium, chloride B Chloride, sodium, magnesium, sulfur C Sulfur, chloride, magnesium, sodium D Sulfur, magnesium, sodium, chloride 4. About how man times greater is the concentration of chloride than the concentration of magnesium? A B 0.66 C.5 D 5 8. The graph of which eponential function is shown? A 5 3 B 53 C 5 p 3 D 5 p 3 9. How does the graph compare with the graph of 5 3? A It is a vertical stretch. B It is a vertical shrink. C It is a reflection in the -ais. D It is the same graph. 550 Chapter 8 Eponents and Eponential Functions