Use Scientific Notation

8.4 Use Scientific Notation Before You used properties of exponents. Now You will read and write numbers in scientific notation. Why? So you can compare lengths of insects, as in Ex. 51. Key Vocabulary scientific notation Numbers such as 1,000,000, 153,000, and 0.0009 are written in standard form. Another way 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 10 n where 1 c < 10 and n is an integer. Number Standard form Scientific notation Two million Five thousandths 2,000,000 0.005 2 3 10 6 5 3 10 23 E XAMPLE 1 Write numbers in scientific notation a. 42,590,000 5 4.259 3 10 7 Move decimal point 7 places to the left. Exponent is 7. b. 0.0000574 5 5.74 3 10 25 Move decimal point 5 places to the right. Exponent is 25. E XAMPLE 2 Write numbers in standard form READING A positive number in scientific notation is greater than 1 if the exponent is positive. A positive number in scientific notation is between 0 and 1 if the exponent is negative. a. 2.0075 3 10 6 5 2,007,500 Exponent is 6. Move decimal point 6 places to the right. b. 1.685 3 10 24 5 0.0001685 Exponent is 24. Move decimal point 4 places to the left. at classzone.com GUIDED PRACTICE for Examples 1 and 2 1. Write the number 539,000 in scientific notation. Then write the number 4.5 3 10 24 in standard form. 512 Chapter 8 Exponents and Exponential Functions

E XAMPLE 3 Order numbers in scientific notation Order 103,400,000, 7.8 3 10 8, and 80,760,000 from least to greatest. Solution STEP 1 Write each number in scientific notation, if necessary. 103,400,000 5 1.034 3 10 8 80,760,000 5 8.076 3 10 7 STEP 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10. Because 10 7 < 10 8, you know that 8.076 3 10 7 is less than both 1.034 3 10 8 and 7.8 3 10 8. Because 1.034 < 7.8, you know that 1.034 3 10 8 is less than 7.8 3 10 8. So, 8.076 3 10 7 < 1.034 3 10 8 < 7.8 3 10 8. STEP 3 Write the original numbers in order from least to greatest. 80,760,000; 103,400,000; 7.8 3 10 8 E XAMPLE 4 Compute with numbers in scientific notation Evaluate the expression. Write your answer in scientific notation. a. (8.5 3 10 2 )(1.7 3 10 6 ) AVOID ERRORS Notice that 14.45 3 10 8 is not written in scientific notation because 14.45 > 10. 5 (8.5 p 1.7) 3 (10 2 p 10 6 ) Commutative property and associative property 5 14.45 3 10 8 Product of powers property 5 (1.445 3 10 1 ) 3 10 8 Write 14.45 in scientific notation. 5 1.445 3 (10 1 3 10 8 ) Associative property 5 1.445 3 10 9 Product of powers property b. (1.5 3 10 23 ) 2 5 1.5 2 3 (10 23 ) 2 Power of a product property 5 2.25 3 10 26 Power of a power property REVIEW FRACTIONS For help with fractions, see p. 915. c. 1.2 3 10 4 1.6 3 10 23 5 1.2 1.6 3 104 10 23 Product rule for fractions 5 0.75 3 10 7 Quotient of powers property 5 (7.5 3 10 21 ) 3 10 7 Write 0.75 in scientific notation. 5 7.5 3 (10 21 3 10 7 ) Associative property 5 7.5 3 10 6 Product of powers property GUIDED PRACTICE for Examples 3 and 4 2. Order 2.7 10 5, 3.401 10 4, and 27,500 from least to greatest. Evaluate the expression. Write your answer in scientific notation. 3. (1.3 3 10 25 ) 2 4.5 3 10 4. 5 5. (1.1 3 10 7 )(4.2 3 10 2 ) 1.5 3 10 22 8.4 Use Scientific Notation 513

E XAMPLE 5 Solve a multi-step problem BLOOD VESSELS Blood flow is partially controlled by the cross-sectional area of the blood vessel through which the blood is traveling. Three types of blood vessels are venules, capillaries, and arterioles. Capillary Venule Arteriole r r r r = 5.0 x 10 3 mm r = 1.0 x 10 2 mm r = 5.0 x 10 1 mm a. Let r 1 be the radius of a venule, and let r 2 be the radius of a capillary. Find the ratio of r 1 to r 2. What does the ratio tell you? b. Let A 1 be the cross-sectional area of a venule, and let A 2 be the cross-sectional area of a capillary. Find the ratio of A 1 to A 2. What does the ratio tell you? 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, you can see that the radius of the venule r 1 is 1.0 3 10 22 millimeter and the radius of the capillary r 2 is 5.0 3 10 23 millimeter. r 1 1.0 3 1022 r2 5 5 1.0 3 1022 5 0.2 3 5.0 3 10 23 5.0 10 23 101 5 2 ANOTHER WAY You can also find the ratio of the crosssectional areas by finding the areas using the values for r 1 and r 2, setting up a ratio, and then simplifying. The ratio tells you that the radius of the venule is twice the radius of the capillary. b. To find the cross-sectional areas, use the formula for the area of a circle. A 1 2 5 πr 1 A2 2 πr2 5 r 2 1 2 r2 Write ratio. Divide numerator and denominator by p. 5 1 r 1 r2 2 2 Power of a quotient property 5 2 2 5 4 Substitute and simplify. The ratio tells you that the cross-sectional area of the venule is four times the cross-sectional area of the capillary. 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 Example 5 6. WHAT IF? Compare the radius and cross-sectional area of an arteriole with the radius and cross-sectional area of a capillary. 514 Chapter 8 Exponents and Exponential Functions

8.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 3, 17, and 53 5 STANDARDIZED TEST PRACTICE Exs. 2, 15, 48, 49, 54, and 59 5 MULTIPLE REPRESENTATIONS Ex. 58 1. VOCABULARY Is 0.5 3 10 6 written in scientific notation? Explain why or why not. 2. WRITING Is 7.89 3 10 6 between 0 and 1 or greater than 1? Explain how you know. EXAMPLE 1 on p. 512 for Exs. 3 15 WRITING IN SCIENTIFIC NOTATION Write the number in scientific notation. 3. 8.5 4. 0.72 5. 82.4 6. 0.005 7. 72,000,000 8. 0.00406 9. 1,065,250 10. 0.000045 11. 1,060,000,000 12. 0.00000526 13. 900,000,000,000,000 14. 0.00000007008 15. MULTIPLE CHOICE Which number represents 54,004,000,000 written in scientific notation? A 54004 3 10 6 B 54.004 3 10 9 C 5.4004 3 10 10 D 0.54004 3 10 11 EXAMPLE 2 on p. 512 for Exs. 16 28 WRITING IN STANDARD FORM Write the number in standard form. 16. 2.6 3 10 3 17. 7.5 3 10 7 18. 1.11 3 10 2 19. 3.03 3 10 4 20. 4.709 3 10 6 21. 1.544 3 10 10 22. 6.1 3 10 23 23. 4.4 3 10 210 24. 2.23 3 10 26 25. 8.52 3 10 28 26. 6.4111 3 10 210 27. 1.2034 3 10 26 28. ERROR ANALYSIS Describe and correct the error in writing 1.24 3 10 23 in standard form. 1.24 3 10 23 5 1240 EXAMPLE 3 on p. 513 for Exs. 29 32 ORDERING NUMBERS Order the numbers from least to greatest. 29. 45,000; 6.7 3 10 3 ; 12,439; 2 3 10 4 30. 65,000,000; 6.2 3 10 6 ; 3.557 3 10 7 ; 55,004,000; 6.07 3 10 6 31. 0.0005; 9.8 3 10 26 ; 5 3 10 23 ; 0.00008; 0.04065; 8.2 3 10 23 32. 0.0000395; 0.00010068; 2.4 3 10 25 ; 5.08 3 10 26 ; 0.000005 COMPARING NUMBERS Copy and complete the statement using <, >, or 5. 33. 5.6 3 10 3? 56,000 34. 404,000.1? 4.04001 3 10 5 35. 9.86 3 10 23? 0.00986 36. 0.003309? 3.309 3 10 23 37. 2.203 3 10 24? 0.0000203 38. 604,589,000? 6.04589 3 10 7 8.4 Use Scientific Notation 515

EXAMPLE 4 on p. 513 for Exs. 39 48 EVALUATING EXPRESSIONS Evaluate the expression. Write your answer in scientific notation. 39. (4.4 3 10 3 )(1.5 3 10 27 ) 40. (7.3 3 10 25 )(5.8 3 10 2 ) 41. (8.1 3 10 24 )(9 3 10 26 ) 42. 6 3 10 23 8 3 10 26 43. 5.4 3 10 25 1.8 3 10 22 44. 4.1 3 10 4 8.2 3 10 8 45. (5 3 10 28 ) 3 46. (7 3 10 25 ) 4 47. (1.4 3 10 3 ) 2 48. MULTIPLE CHOICE Which number is the value of 1.235 3 104 9.5 3 10 7? A 0.13 3 10 24 B 1.3 3 10 24 C 1.3 3 10 23 D 0.13 3 10 3 49. OPEN ENDED Write two numbers in scientific notation whose product is 2.8 3 10 4. Write two numbers in scientific notation whose quotient is 2.8 3 10 4. 50. CHALLENGE Add the numbers 3.6 3 10 5 and 6.7 3 10 4 without writing the numbers in standard form. Write your answer in scientific notation. Describe the steps you take. PROBLEM SOLVING EXAMPLE 3 on p. 513 for Exs. 51 52 51. 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 2.5 3 10 21 Walking stick 555 Parasitic wasp 1.4 3 10 24 Elephant beetle 1.67 3 10 2 52. ASTRONOMY The spacecrafts Voyager 1 and Voyager 2 were launched in 1977 to gather data about our solar system. As of March 12, 2004, Voyager 1 had traveled a total distance of about 9,643,000,000 miles, and Voyager 2 had traveled a total distance of about 9.065 3 10 9 miles. Which spacecraft had traveled the greater distance at that time? EXAMPLE 4 on p. 513 for Ex. 53 EXAMPLE 5 on p. 514 for Exs. 54 55 53. AGRICULTURE In 2002, about 9.7 3 10 8 pounds of cotton were produced in California. The cotton was planted on 6.9 3 10 5 acres of land. What was the average number of pounds of cotton produced per acre? Round your answer to the nearest whole number. 54. SHORT RESPONSE The average flow rate of the Amazon River is about 7.6 3 10 6 cubic feet per second. The average flow rate of the Mississippi River is about 5.53 3 10 5 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 you? 516 5 WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

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 you? 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 you? c. What is the relationship between the ratios of the radii and the ratios of the volumes? 56. MULTI-STEP PROBLEM In 1954, 50 swarms of locusts were observed in Kenya. The largest swarm covered an area of 200 square kilometers. The average number of locusts in a swarm is about 5 3 10 7 locusts per square kilometer. a. About how many locusts were in Kenya s largest swarm? Write your answer in scientific notation. b. The average mass of a desert locust is 2 grams. What was the total mass (in kilograms) of Kenya s largest swarm? Write your answer in scientific notation. 57. DIGITAL PHOTOGRAPHY When a picture is taken with a digital camera, the resulting image is made up of square pixels (the smallest unit that can be displayed on a monitor). For one image, the side length of a pixel is 4 3 10 23 inch. A print of the image measures 1 3 10 3 pixels by 1.5 3 10 3 pixels. What are the dimensions of the print in inches? 58. MULTIPLE REPRESENTATIONS The speed of light is 1.863 3 10 5 miles per second. a. Writing an Expression Assume 1 year is 365 days. Write an expression to convert the speed of light from miles per second to miles per year. b. Making a Table Make a table that shows the distance light travels in 1, 10, 100, 1000, 10,000, and 100,000 years. Our galaxy has a diameter of about 5.875 3 10 17 miles. Based on the table, about how long would it take for light to travel across our galaxy? 59. EXTENDED RESPONSE When a person is at rest, approximately 7 3 10 22 liter of blood flows through the heart with each heartbeat. The human heart beats about 70 times per minute. a. Calculate About how many liters of blood flow through the heart each minute when a person is at rest? b. Estimate There are approximately 5.265 3 10 5 minutes in a year. Use your answer from part (a) to estimate the number of liters of blood that flow through the human heart in 1 year, in 10 years, and in 80 years. Write your answers in scientific notation. c. Explain Are your answers to part (b) underestimates oroverestimates? Explain. 8.4 Use Scientific Notation 517

60. CHALLENGE A solar flare is a sudden eruption of energy in the sun s atmosphere. Solar flares are classified according to their peak X-ray intensity (in watts per meter squared) and are denoted with a capital letter and a number, as shown in the table. For example, a C4 flare has a peak intensity of 4 3 10 26 watt per square meter. Class Bn Cn Mn Xn Peak intensity (w/m 2 ) n 3 10 27 n 3 10 26 n 3 10 25 n 3 10 24 a. In November 2003, a massive X45 solar flare was observed. In April 2004, a C9 flare was observed. How many times greater was the intensity of the X45 flare than that of the C9 flare? b. A solar flare may be accompanied by 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 8.2 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 1.5 3 10 11 meters? MIXED REVIEW PREVIEW Prepare for Lesson 8.5 in Exs. 61 68. Write the percent as a decimal. (p. 916) 61. 33% 62. 62.7% 63. 0.9% 64. 0.04% 65. 3.95% 66. 1 4 % 67. 5 2 % 68. 133% Graph the equation. 69. x 525 (p. 215) 70. y 5 4 (p. 215) 71. 3x 2 7y 5 42 (p. 225) 72. y 2 2x 5 12 (p. 225) 73. y 522x 1 6 (p. 244) 74. y 5 1.5x 2 9 (p. 244) QUIZ for Lessons 8.3 8.4 Simplify the expression. Write your answer using only positive exponents. (p. 503) 1. (24x) 4 p (24) 26 2. (23x 7 y 22 ) 23 3. 1 (5z) 23 4. (6x) 22 y 5 2x 3 y 27 Write the number in standard form. (p. 512) 5. 6.02 3 10 6 6. 5.41 3 10 11 7. 8.007 3 10 25 8. 9.253 3 10 27 9. DINOSAURS The estimated masses of several dinosaurs are shown in the table. (p. 512) 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 77,100 Diplodocus 1.06 3 10 4 Apatosaurus 29,900 Ultrasaurus 1.36 3 10 5 518 EXTRA PRACTICE for Lesson 8.4, p. 945 ONLINE QUIZ at classzone.com

Graphing Calculator ACTIVITY Use after Lesson 8.4 8.4 Use Scientific Notation classzone.com Keystrokes QUESTION How can you use a graphing calculator to solve problems that involve numbers in scientific notation? E XAMPLE Use numbers in scientific notation Gold is one of many trace elements dissolved in seawater. There is about 1.1 3 10 28 gram of gold per kilogram of seawater. The mass of the oceans is about 1.4 3 10 21 kilograms. About how much gold is present in the oceans? STEP 1 Write a verbal model Amount of gold present in oceans (grams) 5 Amount of gold in 1 kilogram of seawater (gram/kilogram) p Amount of seawater in oceans (kilograms) STEP 2 Find product The product is (1.1 3 10 28 ) p (1.4 3 10 21 ). 1.1 10 8 1.4 10 21 STEP 3 Read result The calculator indicates that a number is in scientific notation by using E. You can read the calculator s result 1.54E13 as 1.54 3 10 13. There are about 1.54 3 10 13 grams of gold present in the oceans. (1.1*10^-8)(1.4*10 ^21) 1.54E13 P RACTICE Evaluate the expression. Write the result in scientific notation. 1. (1.5 3 10 4 )(1.8 3 10 9 ) 2. (2.6 3 10 214 )(1.4 3 10 20 ) 3. (7.0 3 10 25 ) 4 (2.8 3 10 6 ) 4. (4.5 3 10 15 ) 4 (9.0 3 10 22 ) 5. GASOLINE A scientist estimates that it takes about 4.45 3 10 7 grams of carbon from ancient plant matter to produce 1 gallon of gasoline. In 2002 motor vehicles in the U.S. used about 1.37 3 10 11 gallons of gasoline. a. If all of the gasoline used in 2002 by motor vehicles in the U.S. came from carbon from ancient plant matter, how many grams of carbon were used to produce the gasoline? b. There are about 5.0 3 10 22 atoms of carbon in 1 gram of carbon. How many atoms of carbon were used? 8.4 Use Scientific Notation 519