Exponent Laws and Scientific Notation
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1 Investigation 4 Exponent Laws and Scientific Notation Lucita and Tala were discussing how to multiply by We can start by rearranging things a little. 4.1 times 3 is 12.3, and we can use a product rule for the powers of But the answer's not in scientific notation. The first number is greater than Rewrite it as Think & Discuss How would you divide two numbers written in scientific notation? For example, what is ( ) ( )? Write your answer in scientific notation. Develop & Understand: A For the following exercises, do not use your calculator unless otherwise indicated. 1. There are about stars in our galaxy and about galaxies in the observable universe. a. Suppose every galaxy has as many stars as ours. How many stars are there in the observable universe? Show how you found your answer. Real-World Link Estimates of the number of stars and the number of galaxies are revised as astronomers gather more data on the universe. b. Suppose only 1 in every 1,000 stars in the observable universe has a planetary system. How many planetary systems are there? Show how you found your answer. c. Suppose 1 in every 1,000 of those planetary systems has at least one planet with conditions suitable for life as we know it. How many such systems are there? Show how you found your answer. d. At the end of the 20th century, the world population was estimated at about 6 billion people. Compare this number to your answer in Part c. What does your answer mean in terms of the situation? 86 Unit E Exponents
2 Math Link You may recall that 1 billion is 1,000,000,000. What is its equivalent in scientific notation? 2. Escherichia coli is a type of bacterium that is sometimes found in -12 swimming pools. Each E. coli bacterium has a mass of 2 10 gram. The number of bacteria increase so that after 30 hours, one 8 bacterium has been replaced by a population of bacteria. a. Suppose a pool begins with a population of only one bacterium. What would be the mass of the population after 30 hours? b. A small paper clip has a mass of about 1 gram. How does the 8 mass of the paper clip compare to the mass of the E. coli bacteria? Show how you found your answer. 3. The speed of light is about miles per second. a. On average, it takes light about 500 seconds to travel from the Sun to Earth. What is the average distance from Earth to the Sun? Write your answer in scientific notation. b. The star Alpha Centauri is approximately miles from Earth. How many seconds does it take light to travel between Alpha Centauri and Earth? c. Use your answer to Part b to estimate how many years it takes for light to travel between Alpha Centauri and Earth. You may use your calculator. Real-World Link There are 88 recognized constellations, or groupings of stars, that form easily identified patterns. This is the constellation Orion, the Hunter. 4. These data show current estimates of the energy released by the three largest earthquakes recorded on Earth. The joule is a unit for measuring energy, named after British physicist James Prescott Joule. Largest Recorded Earthquakes Location Date Energy Released (joules) Tambora, Indonesia April Santorini, Greece about 1470 B.C Krakatoa, Indonesia August a. The Santorini earthquake was how many times as powerful as the Krakatoa earthquake? b. The Tambora earthquake was how many times as powerful as the Krakatoa earthquake? Investigation 4 87
3 5. A scientist is growing a culture of cells. The culture currently contains cells. a. The number of cells doubles every day. If the scientist does not use any of the cells for an experiment today, how many cells will she have tomorrow? b. Suppose she uses of the cells for an experiment. How many will she have left? Show how you found your answer. Be careful. This exercise is different from the others you have done. Develop & Understand: B Copy these multiplication and division charts. Without using your calculator, find the missing expressions. Write all entries using scientific notation. For the division chart, divide the row label by the column label ? ? a 8 10 a Share & Summarize Jordan has written his calculations for three expressions involving scientific notation. Check his work on each exercise. If his work is correct, write correct. If it is incorrect, write a note explaining his mistake and how to correctly solve the exercise. 1. ( ) 2 = = ( ) ( ) = = ( ) - ( ) = = Unit E Exponents
4 Inquiry Investigation 5 Model the Solar System Inquiry Materials masking tape ruler Real-World Link For seventy-five years, Pluto was classified as a planet. However, in 2006, the International Astronomical Union revised the definition of planet. Pluto was reclassified as a dwarf planet. In this investigation, you will examine the relative distances between objects in the solar system. Make a Prediction The average distances of planets from the Sun are often written in scientific notation. Planet Average Distance from Sun (miles) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Imagine lining up the planets with the Sun on one end and Neptune on the other so that each planet is at its average distance from the Sun. How would the planets be spaced? Here is one student s prediction. Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 1. Now make your own prediction. Without using your calculator, sketch a scale version of the planets lined up in a straight line from the sun. Do not worry about the sizes of the planets. Create a Model To check your prediction, some of the members of your class will represent parts of the solar system in a large-scale model. The scale model will allow you to compare the planets average distances from the sun. However, it will not model the relative sizes of the planets. In a large space, use masking tape to mark a line along which you will make your model. The sun will be at one end of the line and Neptune will be at the other. As a class, answer the following questions to determine the locations of the planets Go on in your model. Investigation 5 89
5 Inquiry 2. Measure the length of the line. This will be the distance between the sun and Neptune in your model. 3. What is the actual distance between the sun and Neptune in miles? 4. What number would you multiply distances, in feet, in your model by to find the actual distance, in miles? 5. How can you calculate the distances of the planets from the Sun in your model? 6. Use your answer to Question 5 to estimate the distance between each planet and the Sun in your scale model. Copy the table, fill in the proper unit in the last column, and then record the scaled distances. Real-World Link Many Web sites contain information about the solar system. Sites may use a common convention for scientific notation in which the symbols 10 are replaced by the letter e. The exponent follows as a full-size numeral. Using this convention, is written 2.8 e7. Average Distance Planet from Sun (miles) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Average Distance from Sun in Scale Model (? ) Your teacher will assign your group to a planet or the Sun. Your group should create a sign stating the name of your planet or the Sun and its average distance from the Sun, in miles, in scientific notation. Decide as a class at which end of the line the Sun will be. Determine where your celestial body belongs in the scale model. Choose one member of your group to represent your celestial body by standing on the line with the sign. Check Your Prediction Answer the following questions as a class. 7. Which two planets are closest together? What is the actual distance between them? 8. Which two adjacent planets are farthest apart? Adjacent means next to each other. What is the actual distance between them? 90 Unit E Exponents
6 Inquiry 9. Compare the two distances in Questions 7 and 8. How many times farther apart are the planets in Question 8 than the planets in Question 7? 10. Did your sketch give a reasonably accurate picture of the distances? 11. Was there anything about the spacing of the planets that surprised you? If so, what? 12. Was there anything about the spacing of the planets that did not surprise you? If so, what? a. b. c. What Did You Learn? 13. Draw a scale version of the planets that shows the relative distances between the planets. Do not worry about representing the sizes of the planets, but do your best to get the distances between planets correct. 14. Below are three number lines marked with numbers in scientific notation. Which number line has numbers in the correct places? Investigation 5 91
7 Unit E Practice & Apply 1. Social Studies One of these numbers is in standard notation, and one is in scientific notation. One is the world population in 1750, and the other is the world population in ,000,000 Which number do you think is the world population in 1750? In 1950? Explain your reasoning. 2. For what values of n, if any, will n 2 be equal to or less than 0? 3. For what values of n, if any, will n 3 be equal to or less than 0? Real-World Link In 2006, the United States Census Bureau estimated the population of New York City as residents. If each resident produces four pounds of trash per day, that is about pounds of garbage every year. Given that n represents a positive integer, decide whether each statement is sometimes true, always true, or never true. In Exercises 4 7, if a statement is sometimes true, state for what values it is true. 4.4 n = 65, n is less than 1,000,000 (that is, 4 n < 1,000,000) 6. n 2 is negative (that is, n 2 < 0) n is greater than or equal to 0 and, at the same time, 0.9 n is less than or equal to 1. That is, n For what positive values of x will x 20 be greater than x 18? 9. For what positive values of x will x 18 be greater than x 20? 10. For what positive values of x will x 18 be equal to x 20? 11. For what negative values of x will x 20 be greater than x 18? 12. For what negative values of x will x 18 be greater than x 20? 13. For what negative values of x will x 18 be equal to x 20? 92 Unit E Exponents
8 14. Challenge In Investigation 1, you explored positive integer powers of 2 and of 4. n n n ,024 4,096 16,384 65, ,144 Now think about positive integer powers of 8 a. List the first five positive integer powers of 8. b. Name three numbers that are on all three lists, that is, three numbers that are powers of 2, 4, and 8. c. List three numbers greater than 16 that are powers of 2 but are not powers of 8. d. List three numbers greater than 16 that are powers of 4 but are not powers of 8. e. Describe the powers of 2 that are also powers of 8. f. Describe the powers of 4 that are also powers of For what positive values of x will x -20 be greater than x -18? 16. For what positive values of x will x -18 be greater than x -20? 17. For what values, positive or negative, of x will x -18 be equal to x -20? 18. The sixth power of 2 is 64, or 2 6 = 64. a. Write at least five other expressions, using a single base and a single exponent, that are equivalent to 64. b. Write the number 64 using scientific notation. Sort each set of expressions into groups so that the expressions in each group are equal to one another. Do not use your calculator. 19. m 2 ( 1 _ m ) 2 m -2 ( 1 _ m ) -2 1_ m 2 1 m x ( 1 _ 3) x ( 1 _ 3) -x 1_ 3 x 3 -x 1 3 x _ c = ( _ a 21. Prove It! Prove that the second quotient law, a c b b) c, works for positive integer exponents c. Assume b is not equal to 0. On Your Own Exercises 93
9 22. Challenge Prove that the power of a power law, ( a b ) c = a bc, works for positive integer exponents b and c. Rewrite each expression using a single base and a single exponent x 24. (- 4 m ) m (-3) (-3) _ -8 9 ( 28. _ m 84 x ) m n a n a_ ( ) Simplify each expression a 4 3 a m -3 m 4 b (4 x -2 6 ) 10 n -15 _ 5 n (- m 2 n 3 4 ) 37. ( a m ) n ( b 3 2 ) 38. ( x -2 ) 3 x 5 _ b 5 4 b ( x 4 y -5-3 _) (xy) 2 Copy each chart. Without using your calculator, find the missing expressions. Write all entries as powers or products of powers. For the division chart, divide the row label by the column label x - 2 x? x -4 x n a-x?2 2 2a a (2n) 4 a 4 7 Real-World Link Sirius, also called the Dog Star, is a double star orbited by a smaller star called Sirius B, or the Pup. 43. Physical Science The speed of light is about miles per second. At approximately miles from Earth, Sirius appears to be the brightest of the stars. How many seconds does it take light to travel between Sirius and Earth? How many years does it take? 44. Social Studies The population of the world in the year 1 A.D. has been estimated at 200,000,000. By 1850, this estimate had grown to 1 billion. By 2000, the population was close to a. The 1850 population was how many times the 1 A.D. population? b. The 2000 population was how many times the 1850 population? c. Did the world population grow more during the 1,850 years from 1 A.D. to 1850 or during the 150 years from 1850 to 2000? 94 Unit E Exponents
10 45. Copy this division chart. Without using your calculator, find the missing expressions by dividing the row label by the column label. Express all entries in scientific notation.?? x ? a-x-1 Connect & Extend 46. Social Studies According to the 1790 census, the population of the United States was 3,929,214. You can approximate this value with powers of various numbers. For example, 2 21 is 2,097,152 and 2 22 is 4,194,304. Using powers of 2, the number 2 22 is the closest possible approximation to 3,929,214. What is the closest possible approximation using powers of 3? Powers of 4? Powers of 5? 47. Which of these sets of numbers share numbers with the powers of 2? Explain how you know. a. positive integer powers of 6 b. positive integer powers of 7 c. positive integer powers of Fine Arts A piano has eight C keys. The frequency of a note determines how high or low it sounds. Moving from the left of the keyboard to the right, each C note has twice the frequency of the one before it. For example, middle C has a frequency of about vibrations per second. The next higher C has a frequency of about vibrations per second. If the first C key has a frequency of x, what is the frequency of the last C key? C D E F G A B C D E F On Your Own Exercises 95
11 49. Economics Julián s mother offered him $50 a month in allowance. Julián said he would rather have his mother pay him 1 penny the first day of the month, 2 pennies the second day, 4 the third day, 8 the fourth day, and so on. His mother would simply double the number of pennies she gave him each day until the end of the month. His mother said that sounded fine with her. a. Would Julián receive more money with an allowance of $50 a month or using his plan? Explain why. b. If Julián s plan produces more money, on what day would he receive more than $50 a month? c. With his plan, how much money would Julián receive the last day of June, which has 30 days? d. Challenge With his plan, how much would Julián receive in all for the month of June? Filling in a table like the one below might help you answer this question. Day Amount Received Each Day Total Amount 1 $0.01 $ $0.02 $ = $ $0.04 $ = $ A particular tennis tournament begins with 64 players. If a player loses a single match, he or she is knocked out of the tournament. After one round, only 32 players remain. After two rounds, only 16 players remain, and so on. Six students have conjectured a formula to describe the number of players remaining p after r rounds. Which rule or rules are correct? For each rule you think is correct, show how you know. 64 Tessa: p = _ r 2 Peter: p = 64 (_ 1 2) r -r Marla: p = 64 2 Damon: p = r Antonia: p = 64 _ 1 r r 2 Tamera: p = 64 (-2) 96 Unit E Exponents
12 51. This list of numbers continues in the same pattern in both directions...., _ 1, 1, 5, 25, 125, 625,... 5 Ivana wanted to write an expression for this list using n as a variable. To do that, she had to choose a number on the list to be her starting point. She decided that when n = 1, the number on the list is 5. When n = 2, the number is 25. a. Using Ivana s plan, write an expression that will give any number on the list. b. What value for n gives you 625? 1? 1 _ 5? Without computing the value of each pair of numbers, determine which number is greater. For each exercise, explain why or ,600 or or A pastry shop sells a square cake that is 45 cm wide and 10 cm thick. A competitor offers a square cake of the same thickness that is 2 cm wider. The first baker argues that the area of the top of the rival cake is ( ) 2 cm 2 and is therefore only 4 c m 2 larger than the one he sells. What computational mistake did the first baker make? What is the actual difference in areas? Astronomy Earth travels around the sun approximately 6 10 miles each year. At approximately what speed must Earth travel in miles per second? Give your answer in scientific notation. 97
13 57. Life Science The diameter of the body of a Purkinje cell is m. a. If a microscope magnifies 1,000 times, what will be the scaled diameter, in meters, as viewed in the microscope? b. What is the scaled diameter, in centimeters, as viewed in the microscope? 58. In Your Own Words Write a letter to a student who is confused about exponents. Explain how to multiply two numbers when each is written as a base to an exponent. Be sure to address the following. The numbers have the same exponent. The numbers have the same base. The numbers have different exponents and different bases. 98 Unit E Exponents
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