Topic 6: Exponents and Scientific Notation Scientific notation is a short way to write very large or very small numbers. Scientific notation uses powers of 10 and exponents. Getting Ready for Problem 6.1 In scientific notation a number is expressed as a product, where the first factor is a number greater than or equal to 1 but less than 10, and the second factor is a power of 10. Here are different ways to write the same number: • How can the exponential form of a number help you write a number in scientific notation? • Compare scientific notation to the standard form. How does the decimal point move? • Describe how to write 0.0025 in scientific notation. The Bayside School Science Club often goes on field trips related to their science projects. Club members often have to work with very large or very small numbers to solve problems. Standard form 145,000 Expanded form 100,000 40,000 5,000 Exponential form 1 10 5 4 10 4 5 10 3 Scientific notation 1.45 10 5
Transcript
Topic 6: Exponents and Scientific Notation
Scientific notation is a short way to write very large or very small numbers.Scientific notation uses powers of 10 and exponents.
Getting Ready for Problem 6.1
In scientific notation a number is expressed as a product, where the firstfactor is a number greater than or equal to 1 but less than 10, and thesecond factor is a power of 10.
Here are different ways to write the same number:
• How can the exponential form of a number help you write a number inscientific notation?
• Compare scientific notation to the standard form. How does the decimalpoint move?
• Describe how to write 0.0025 in scientific notation.
The Bayside School Science Club often goes on field trips related to theirscience projects. Club members often have to work with very large or verysmall numbers to solve problems.
Standard form 145,000
Expanded form 100,000 � 40,000 � 5,000
Exponential form 1 � 105 � 4 � 104 � 5 � 103
Scientific notation 1.45 � 105
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On a field trip to an observatory, some of the club members noticed thistable, displayed near one of the telescopes.
A. The club members decide to write the diameter of each planet usingscientific notation.
1. Write the diameters of Uranus and Venus in scientific notation.
2. Jackie writes the diameter of Jupiter as 8.8846 � 104 miles. Is Jackiecorrect? Explain.
3. Why do you think the planet diameters are given in standard formand the average distances are given in scientific notation?
B. Margaret claims that Saturn’s average distance from the Sun is greaterthan Neptune’s average distance from the Sun because the number forSaturn starts with 8 and the number for Neptune starts with 2. Is shecorrect? Explain.
C. Alpha Centauri is the closest star system to our solar system. It isabout 9.2 � 103 times farther from the Sun than Neptune. Estimatehow far Alpha Centauri is from the Sun.
Problem 6.1
PLANET QUICK FACTS
Diameter Average Distance from the SunPlanet (in miles) (to the nearest 10,000 miles)
Mercury 3,032 3.599 � 107
Venus 7,521 6.723 � 107
Earth 7,926 9.296 � 107
Mars 4,222 1.4164 � 108
Jupiter 88,846 4.8363 � 108
Saturn 74,898 8.8819 � 108
Uranus 31,763 1.78396 � 109
Neptune 30,778 2.79884 � 109
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Chris and Aamna are preparing a presentation on very small objects for thenext club meeting.
A. Chris has found some information about atoms. Aamna is using amicroscope to observe some microscopic objects. Some informationthey found appear in the tables below.
1. Why might Chris want to write the particle masses in scientificnotation?
2. Write the particle masses in scientific notation.
B. Aamna is listing the microscopic objects in standard form from least togreatest width.
1. When she converts the widths from scientific notation to standardform, in which direction does she move the decimal point?
2. List the microscopic objects in order from least to greatest widthusing standard notation.
C. Aamna decides to compare the diameter of a pinhead to the width of a small bacterium. The diameter of a pinhead is about 2 × 10–1 cm.Describe how she can estimate the number of small bacteria that couldfit across a pinhead.
Problem 6.2
Building Blocks of Atoms
Particle Mass (mg)
Proton 0.000000000000000000001673
Electron 0.0000000000000000000000009109
Neutron 0.000000000000000000001675
Microscope Observations
Microscopic Object Width (cm)
Human Blood Cell l � 10–3
Small Bacterium l � 10–4
Large Bacterium 7.5 � 10–2
Onion Skin Cell 4 � 10–3
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Problem 6.3
To solve a problem, two of the Science Club members need to simplifyexpressions with exponents.
A. Luanne wants to find a way to simplify 32 � 35. She rewrites the twofactors as 3 � 3 and 3 � 3 � 3 � 3 � 3.
1. How many times does the factor 3 appear in the product?
2. How is this number related to the two exponents?
3. Combine the expressions using one exponent.
4. What did you notice about what happens to exponents when youmultiply numbers with the same base?
B. Luanne uses the same strategy to see what happens to the exponents
when she simplifies the expression . Luanne rewrites the expression
as . Write a general rule for what happens to
exponents when you divide numbers with the same base.
C. Teymour wants to find a way to simplify the expression (23)2. Using asimilar method that Luanne used, Teymour rewrites the expression 23
as 2 � 2 � 2.
1. The exponent 2 in the original expression means that the expressioninside the parentheses must be multiplied by itself. How willTeymour apply this to his expression?
2. Look at the number of factors that Teymour has now. Write anumber in exponential form to represent all of the factors.
3. What did you notice about what happens to exponents when youraise them to a power?
3 3 3 3 3 3 3 3 33 3 3 3 3
35
33
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ExercisesFor Exercises 1–4, use the table below.
1. Write the length of the coastline of the Arctic Ocean in scientificnotation.
2. Write the area of Lake Superior in standard form.
3. Write the length of the coastline of the Pacific Ocean in scientificnotation.
4. The area of the Atlantic Ocean is about 4 � 104 times as great as thearea of Lake Okeechobee. Write the approximate area of the AtlanticOcean in scientific notation.
Body of Water Information
Area Coastline/ShorelineBody of Water (in square km) (to the nearest hundred km)
Arctic Ocean 1.4056 � 107 45,400
Lake Okeechobee 1.89 � 103 200
Lake Superior 8.21 � 104 4,400
Pacific Ocean 1.55557 � 108 135,700
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For Exercises 5–8, use the table below.
5. Write the width of a proton in scientific notation.
6. Write the width of a virus in standard form.
7. Write the diameter of an H2O molecule in scientific notation.
8. A football field is about 1 × 102 m long. About how many dustparticles, laid end to end, would it take to equal the length of a football field?
Very Small Objects
Object Width (m)
Proton 0.0000000000000000031
H2O Molecule 0.00000000035
Virus 1 × 10–7
Small Dust Particle 2 × 10–4
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For Exercises 9–16, use the laws of exponents below.
am � an � am � n (am)n � am � n am � bm � (ab)m
9. Jill says 22 � 33 � 65. Is she correct? Explain.
10. Explain why each of the following statements is true.
a. b.
11. The table shows the number of points some cards are worth in thegame Exponential Frenzy.
Write the value of each card using a positive exponent.
12. Multiple Choice Which of the following is equal to (129)3?
A. 1227 B. 1212 C. 126 D. 123
13. Multiple Choice Which of the following is equal to ?
A. B. 10�4 C. 104 D. 1051104
11024
53
55 55 3 5 3 5
5 3 5 3 5 3 5 3 5 5152
53
55 5 522
am
an 5 am2n
Exponential Frenzy Card Values
Card Number of Points
A 3�2 × 2�2
B
C (4�2)�3
D 5�12 � 510
228
24
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14. Simplify each expression. Write using positive exponents.
a. 154 � 1522 b. c. (35)�6 d. 43 � 53
15. Explain why is a true statement.
16. a. Simplify the expression using the laws of exponents.
b. Evaluate the expression. Write your answer in simplest form.
17. The age of the Universe is about 13,700,000,000 years. Earth’s age is about 4,550,000,000 years.
a. Write each number in scientific notation.
b. Estimate Earth’s age as a fraction of the age of the Universe.
18. Spaceship Earth at Epcot Center in Orlando, Florida, is roughly theshape of a sphere. Use the information below to estimate the volumeof Spaceship Earth. Use 3.14 for π.
19. A sheet of paper is about 0.0032 inch thick. If you start with one sheetof paper, and then double the number of sheets repeatedly, continuinguntil you’ve doubled it 10 times, how thick will your stack of paper be?
223554
3252
723
72 5175
89
84
Formula for the Approximate Radius ofVolume of a Sphere Spaceship Earth
(2.086)6 feetV 543pr3
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Topic 6: Exponents and Scientific Notation
Guided Instruction
Mathematical Goals
• Use scientific notation and exponents to work with and solve problemsinvolving large and small numbers.
• Use laws of exponents to simplify expressions.
Vocabulary
• scientific notation
At a Glance
In this topic, students write large and small numbers using scientific notation,and write numbers expressed in scientific notation in standard form. Theyexplore and discover the laws of exponents, and use those laws to simplifyexpressions with exponents.
In the first two problems, students are introduced to numbers that areeasier to work with when written in scientific notation. They also explore theneed for numbers in scientific notation to be written in standard form. Thethird problem provides students the opportunity to understand methods toquickly simplify expressions involving exponents and to write their own rulesfor doing so.
Use the introduction to have students discuss situations where they haveencountered very large or very small numbers.
Problem 6.1
Before Problem 6.1, during Getting Ready, make sure studentsunderstand how to write a number in scientific notation. Then ask:
• Compare the exponential form of 145,000 and the same number writtenin scientific notation. What do you notice? (Sample answer: The firstterm helps you find what power of 10 to use.)
• Why is 1.45 used as the first factor? (It is between 1 and 10.)
• What steps would you take to write 2,500 using scientific notation, andhow does the decimal point move? (Sample answer: I can write 2,500 as2.5 � 103. The decimal point moves three digits to the left.)
During Problem 6.1 B, ask: When comparing numbers in scientificnotation, what parts of the numbers should you compare first? Why? (Theexponents in the powers of 10; the greater the exponent, the greater thenumber.)
During Problem 6.1 C, ask:
• What operation can you use to solve this problem? (Multiplication)Remind students that they can round the decimal parts of each factorbefore multiplying.
PACING 3 days
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Problem 6.2
During Problem 6.2 A, Part 2, ask: How can you find the correct exponentto use with the power of 10? (Sample answer: Count the number of placesfrom the decimal point to the first non-zero digit.)
During Problem 6.2 C, if students are having difficulty determining whichoperation to use, ask: If you were trying to find the number of 2-foot-longbrick pavers that would fit across the middle of a circular driveway with adiameter of 20 feet, which operation would you use? (Division)
Problem 6.3
Before Problem 6.3 A, ask: For 82, which is the base and which is theexponent? (8 is the base and 2 is the exponent.)
During Problem 6.3 A, Part 1, ask: What does the number of times that 3appears as a factor tell you? (It shows what exponent to use if I were towrite the repeated multiplication using exponential notation.)
Before Problem 6.3 B, ask: What is an easy way to simplify ?
(Sample answer: , so one 5 in the numerator and the 5 in the
denominator “cancel” each other, since 1 multiplied by any number
is that number. So, � 5 � 1 � 5.)
During Problem 6.3 B, ask: To write a general rule, would you use wordsor variables? Explain. (Sample answer: Variables, because I can easilysubstitute the base and exponents and simplify the expression)
Before Problem 6.3 C, Part 1, ask: How could you write (7a)2 using amultiplication symbol? (7a � 7a)
Summarize To summarize the lesson, ask:
• What are some reasons you would write large or small numbers usingscientific notation? (Sample answer: It makes the numbers easier towrite and compute with, and it takes up less space.)
• What procedures can you use to multiply or divide numbers inexponential notation that have the same base? (Multiply: add theexponents and keep the same base. Divide: subtract the exponent inthe denominator from the exponent in the numerator and keep thesame base.)
• What procedure can you use to simplify (x2)5? (Multiply the exponentsand keep the same base.)
You will find additional work on exponents in the CMP2 Unit Growing,Growing, Growing.
