Download Topic 6: Exponents and Scientific Notation

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:
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
• 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.
Problem 6.1
On a field trip to an observatory, some of the club members noticed this
table, displayed near one of the telescopes.
PLANET QUICK FACTS
Planet
Diameter
(in miles)
Average Distance from the Sun
(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
A. The club members decide to write the diameter of each planet using
scientific 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 Jackie
correct? Explain.
3. Why do you think the planet diameters are given in standard form
and the average distances are given in scientific notation?
B. Margaret claims that Saturn’s average distance from the Sun is greater
than Neptune’s average distance from the Sun because the number for
Saturn starts with 8 and the number for Neptune starts with 2. Is she
correct? Explain.
C. Alpha Centauri is the closest star system to our solar system. It is
about 9.2 103 times farther from the Sun than Neptune. Estimate
how far Alpha Centauri is from the Sun.
Problem 6.2
Chris and Aamna are preparing a presentation on very small objects for the
next club meeting.
A. Chris has found some information about atoms. Aamna is using a
microscope to observe some microscopic objects. Some information
they found appear in the tables below.
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
1. Why might Chris want to write the particle masses in scientific
notation?
2. Write the particle masses in scientific notation.
B. Aamna is listing the microscopic objects in standard form from least to
greatest width.
1. When she converts the widths from scientific notation to standard
form, in which direction does she move the decimal point?
2. List the microscopic objects in order from least to greatest width
using 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 could
fit across a pinhead.
Problem 6.3
To solve a problem, two of the Science Club members need to simplify
expressions with exponents.
A. Luanne wants to find a way to simplify 32 35. She rewrites the two
factors 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 you
multiply numbers with the same base?
B. Luanne uses the same strategy to see what happens to the exponents
5
when she simplifies the expression 33 . Luanne rewrites the expression
3
3
3
3
3
3
3
3
3
3
as
. Write a general rule for what happens to
33333
exponents when you divide numbers with the same base.
C. Teymour wants to find a way to simplify the expression (23)2. Using a
similar method that Luanne used, Teymour rewrites the expression 23
as 2 2 2.
1. The exponent 2 in the original expression means that the expression
inside the parentheses must be multiplied by itself. How will
Teymour apply this to his expression?
2. Look at the number of factors that Teymour has now. Write a
number in exponential form to represent all of the factors.
3. What did you notice about what happens to exponents when you
raise them to a power?
Exercises
For Exercises 1–4, use the table below.
Body of Water Information
Body of Water
Area
(in square km)
Coastline/Shoreline
(to the nearest hundred km)
1.4056 107
45,400
Lake Okeechobee
1.89 103
200
Lake Superior
8.21 104
4,400
Pacific Ocean
108
135,700
Arctic Ocean
1.55557 1. Write the length of the coastline of the Arctic Ocean in scientific
notation.
2. Write the area of Lake Superior in standard form.
3. Write the length of the coastline of the Pacific Ocean in scientific
notation.
4. The area of the Atlantic Ocean is about 4 104 times as great as the
area of Lake Okeechobee. Write the approximate area of the Atlantic
Ocean in scientific notation.
For Exercises 5–8, use the table below.
Very Small Objects
Object
Width (m)
Proton
0.0000000000000000031
H2O Molecule
0.00000000035
Virus
1 × 10–7
Small Dust Particle
2 × 10–4
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 dust
particles, laid end to end, would it take to equal the length of
a football field?
For Exercises 9–16, use the laws of exponents below.
am an am n
am
an
5 am2n
(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.
3
a. 55 5 5 22
5
3
53535
b. 55 5
5 12
535353535
5
5
11. The table shows the number of points some cards are worth in the
game Exponential Frenzy.
Exponential Frenzy Card Values
Card
Number of Points
A
32 × 22
B
2 28
24
C
(42)3
D
512 510
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
13. Multiple Choice Which of the following is equal to
A. 1 4
10
B. 104
C. 104
D. 123
1 ?
10 24
D. 105
14. Simplify each expression. Write using positive exponents.
a. 154 1522
9
b. 84
8
c. (35)6
d. 43 53
23
15. Explain why 7 2 5 15 is a true statement.
7
7
2 5 4
16. a. Simplify the expression 2 32 52 using the laws of exponents.
3 5
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 the
shape of a sphere. Use the information below to estimate the volume
of Spaceship Earth. Use 3.14 for π.
Formula for the
Volume of a Sphere
Approximate Radius of
Spaceship Earth
V 5 4 pr3
3
(2.086)6 feet
19. A sheet of paper is about 0.0032 inch thick. If you start with one sheet
of paper, and then double the number of sheets repeatedly, continuing
until you’ve doubled it 10 times, how thick will your stack of paper be?
At a Glance
Topic 6: Exponents and Scientific Notation
PACING 3 days
Mathematical Goals
• Use scientific notation and exponents to work with and solve problems
involving large and small numbers.
• Use laws of exponents to simplify expressions.
Guided Instruction
In this topic, students write large and small numbers using scientific notation,
and write numbers expressed in scientific notation in standard form. They
explore and discover the laws of exponents, and use those laws to simplify
expressions with exponents.
In the first two problems, students are introduced to numbers that are
easier to work with when written in scientific notation. They also explore the
need for numbers in scientific notation to be written in standard form. The
third problem provides students the opportunity to understand methods to
quickly simplify expressions involving exponents and to write their own rules
for doing so.
Use the introduction to have students discuss situations where they have
encountered very large or very small numbers.
Problem 6.1
Before Problem 6.1, during Getting Ready, make sure students
understand how to write a number in scientific notation. Then ask:
• Compare the exponential form of 145,000 and the same number written
in scientific notation. What do you notice? (Sample answer: The first
term 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, and
how does the decimal point move? (Sample answer: I can write 2,500 as
2.5 103. The decimal point moves three digits to the left.)
During Problem 6.1 B, ask: When comparing numbers in scientific
notation, what parts of the numbers should you compare first? Why? (The
exponents in the powers of 10; the greater the exponent, the greater the
number.)
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 factor
before multiplying.
Vocabulary
•
scientific
notation
Problem 6.2
During Problem 6.2 A, Part 2, ask: How can you find the correct exponent
to use with the power of 10? (Sample answer: Count the number of places
from the decimal point to the first non-zero digit.)
During Problem 6.2 C, if students are having difficulty determining which
operation to use, ask: If you were trying to find the number of 2-foot-long
brick pavers that would fit across the middle of a circular driveway with a
diameter 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 the
exponent? (8 is the base and 2 is the exponent.)
During Problem 6.3 A, Part 1, ask: What does the number of times that 3
appears as a factor tell you? (It shows what exponent to use if I were to
write the repeated multiplication using exponential notation.)
55
Before Problem 6.3 B, ask: What is an easy way to simplify
?
5
5
(Sample answer: 1, so one 5 in the numerator and the 5 in the
5
denominator “cancel” each other, since 1 multiplied by any number
55
5 5
is that number. So,
5 1 5.)
5
5
During Problem 6.3 B, ask: To write a general rule, would you use words
or variables? Explain. (Sample answer: Variables, because I can easily
substitute the base and exponents and simplify the expression)
Before Problem 6.3 C, Part 1, ask: How could you write (7a)2 using a
multiplication symbol? (7a 7a)
Summarize To summarize the lesson, ask:
• What are some reasons you would write large or small numbers using
scientific notation? (Sample answer: It makes the numbers easier to
write and compute with, and it takes up less space.)
• What procedures can you use to multiply or divide numbers in
exponential notation that have the same base? (Multiply: add the
exponents and keep the same base. Divide: subtract the exponent in
the denominator from the exponent in the numerator and keep the
same base.)
• What procedure can you use to simplify (x2)5? (Multiply the exponents
and keep the same base.)
You will find additional work on exponents in the CMP2 Unit Growing,
Growing, Growing.
Assignment Guide for Topic 6
Problem 6.1, Exercises 1–4, 17
Problem 6.2, Exercises 5–8
Problem 6.3, Exercises 9–16, 17–19
Answers to Topic 6
Problem 6.1
A. 1. Uranus: 3.1763 104; Venus: 7.521 103
2. Yes, the decimal point moves 4 places to
the left.
3. The average distances are much greater
numbers than the diameters.
B. No. Neptune is at a greater distance because
the exponent for Neptune is 109, which is
greater than 108.
C. (2.79884 109) (9.2 103) (2.8 109) (9 103), or about
2.5 1013 miles.
Problem 6.2
A. 1. Sample answer: The numbers are very
long because of the large number of
zeros; it’s easy to make a mistake when
writing so many zeros.
2. Proton: 1.673 1021 mg;
electron: 9.109 1025 mg;
neutron: 1.675 1021 mg.
B. 1. To the left
2. Small bacterium: 0.0001 cm; human blood
cell: 0.001 cm; onion skin cell: 0.004 cm;
large bacterium: 0.075 cm
C. She can divide the diameter of the pinhead
by the width of a small bacterium:
(2 101) (l 104) 0.2 0.0001 2,000
Problem 6.3
A. 1. 7 times
2. It is the sum of the two exponents.
3. 37
4. You can add the exponents of the factors
to find the exponent of the product.
B. You subtract the exponent of the divisor
from the exponent of the dividend to find
the exponent of the quotient.
C. 1. He can rewrite (23)2 as
(2 2 2) (2 2 2)
2. 26
3. When an exponent is raised to a power,
the exponent of the result is the product
of the original exponents.
Exercises
1. 4.54 104 km
12. A
13. C
2. 82,100 km2
3. 1.357 105
14. a. 1526
km
4. About 8 107
5. 3.1 m
10–18
km2
8. 5 105 or 500,000
9. 22 33 65 is incorrect; 22 33 4 27
108, 65 7,776. You can only add the
exponents when the bases are the same.
53
5(35) ; the exponent of the quotient
55
is the difference of the exponents of the
dividend and the divisor.
53
555
55555
55
5
5
5
1
5
5
5
55
111
1
1
2
2
5
5
1
1
; Card B: 12 ;
2
6
2
1
Card C: 46; Card D: 2
5
11. Card A:
am
a(mn)
an
1
7 5 5 .
7
15. Sample answer: Using
7. 3.5 1010 m
b.
c. 330
d. 203
6. 0.0000001 m
10. a.
b. 85
7 3
7 32
72
16. a.
223554
223352 33102
3252
b. 2,700
17. a. 1.37 1010, 4.55 109
1
3
18. About 1.1 106 ft3
b. About
19. About 3.28 in.