ACTIVITY 4.1 Applying the Product Rules of Powers to Scientific Notation Before a recent class trip to a lake, Vanessa said she wanted to bring back 3 million grains of sand for a classmate who could not go on the trip. If 1 grain of sand weighs approximately 6.7 3 10 27 kg, how heavy are 3 million grains of sand? WORKED EXAMPLE To calculate the total weight of 3,000,000 grains of sand, you would multiply the weight of 1 grain of sand by the total number of grains 0.00000067 kilograms of sand, or (3,000,000 grains of sand). 1 grain of sand You can use scientific notation and the Product Rules of Powers to compute this product more efficiently. Begin by writing the numbers in scientific notation. (6.7 3 10 27 )(3 3 10 6 ) Apply the Associative and Commutative Properties of Multiplication. Apply the Product Rule of Powers to powers with the same base. (6.7 3 3)(10 27 3 10 6 ) (6.7 3 3)(10 2716 ) Simplify each factor of the product. (20.1)(10 21 ) Rewrite in standard form, if desired. 2.01 kg Three million grains of sand weigh approximately 2.01 kilograms. 1. Suppose that a student has lost 15,000 eyelashes in his life. Use scientific notation to calculate the total weight, in kilograms, of his lost eyelashes. Write your answer in scientific notation. LESSON 4: How Much Larger? M5-63
2. An ecologist estimates that it takes approximately 196,000 pounds of buried plant matter to produce one gallon of gasoline. Some energy experts estimate that the United States consumed about 140.4 billion gallons of gasoline in 2015. Calculate the amount of buried plant matter needed to produce the amount of gasoline consumed in 2015. Write your answer in scientific notation. 3. An oil tanker is approximately 1400 feet long. How far would 9500 oil tankers span if they were placed end to end? Remember: A number written in scientific notation has a mantissa greater than or equal to 1 and less than 10. a. Calculate the approximate length of 9500 oil tankers. Write your answer in scientific notation. b. What additional step is required to calculate the answer in Question 3, part (a) that was not required in Question 2? 4. Calculate each product. Express each product in scientific notation. a. (3 3 10 5 )(2 3 10 6 ) b. (9 3 10 4 )(1 3 10 7 ) c. (4.0 3 10 8 )(2.7 3 10 4 ) d. (5.6 3 10 26 )(3.5 3 10 15 ) M5-64 TOPIC 1: Exponents and Scientific Notation
5. Determine the unknown factors in each equation. Explain your reasoning. a. (4 3 10 7 )(? 3?) 5 8 3 10 12 b. (? 3?)(5 3 10 3 ) 5 3.5 3 10 8 ACTIVITY 4.2 Applying the Quotient Rule of Powers to Scientific Notation The Scoville scale measures the hotness of a chili pepper by the amount of capsaicin it contains. Capsaicin is the chemical that puts the heat in chili peppers. The number of Scoville heat units (SHU) indicates the amount of capsaicin present in the food. The table represents the Scoville rating for a variety of peppers. Scoville Rating Type 15,000,000 to 16,000,000 Pure Capsaicin I wonder how much hotter a habanero pepper is than a jalapeño pepper. How could you compare the peppers? 5,000,000 to 5,300,000 Law Enforcement Grade Pepper Spray 350,000 to 570,000 Red Savina Habanero 200,000 to 300,000 Habanero 70,000 to 80,000 Thai Pepper 30,000 to 50,000 Cayenne Pepper, Tabasco Pepper, some Chipotle Peppers 2500 to 8000 Jalapeño Peppers, Paprika (Hungarian) 500 to 2500 Anaheim Pepper (Mild Chile Pepper) 100 to 500 Pimento, Pepperoncini 0 No Heat, Bell Pepper *Source: Mojave Pepper Farm s Pepper Scale LESSON 4: How Much Larger? M5-65
Use the values from the table to answer each question. 1. How many times hotter is the mildest law enforcement grade pepper spray than the hottest pepperoncini? a. First, write a ratio using the values in the table as they appear. b. Next, write your ratio in scientific notation. c. Finally, rewrite your expression using the Quotient Rule of Powers. Explain your reasoning. d. What does your result represent? 2. Answer each pepper heat comparison question. a. How many times hotter is the hottest Thai pepper than the hottest jalapeño pepper? b. How many times hotter is the mildest law enforcement grade pepper spray than the hottest cayenne pepper? 3. What do you notice about the results in Questions 1 and 2? Explain why this happened. M5-66 TOPIC 1: Exponents and Scientific Notation
4. Answer each pepper heat comparison question. a. How many times hotter is the hottest red savina habanero than the mildest jalapeño pepper? b. How many times hotter is the hottest form of capsaicin than the mildest Anaheim pepper? c. How many times hotter is the hottest cayenne pepper than the mildest jalapeño pepper? d. How are these comparisons different from the comparisons in Questions 1 and 2? 5. Determine each quotient. Express the quotient in scientific notation. a. (6 3 108 ) (2 3 10 3 ) b. (9 3 105 ) (3 3 10 9 ) c. (8 3 1024 ) (2 3 10 3 ) LESSON 4: How Much Larger? M5-67