Download 8.4Use Scientific Notation

8.4
Use Scientific Notation
You used properties of exponents.
Before
You will read and write numbers in scientific notation.
Now
So you can compare lengths of insects, as in Ex. 51.
Why?
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.
For Your Notebook
KEY CONCEPT
Scientific Notation
A number is written in scientific notation when it is of the form c 3 10n
where 1 ≤ c < 10 and n is an integer.
Number
Standard form
Scientific notation
Two million
2,000,000
2 3 106
Five thousandths
0.005
5 3 1023
EXAMPLE 1
a. 42,590,000 5 4.259 3 107
Move decimal point 7 places to the left.
Exponent is 7.
b. 0.0000574 5 5.74 3 1025
Move decimal point 5 places to the right.
Exponent is 25.
EXAMPLE 2
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.
✓
Write numbers in scientific notation
Write numbers in standard form
a. 2.0075 3 106 5 2,007,500
Exponent is 6.
Move decimal point 6 places to the right.
b. 1.685 3 1024 5 0.0001685
Exponent is 24.
Move decimal point 4 places to the left.
"MHFCSB
GUIDED PRACTICE
at classzone.com
for Examples 1 and 2
1. Write the number 539,000 in scientific notation. Then write the number
4.5 3 1024 in standard form.
512
Chapter 8 Exponents and Exponential Functions
EXAMPLE 3
Order numbers in scientific notation
Order 103,400,000, 7.8 3 108, 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 3 108
80,760,000 5 8.076 3 107
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 107 < 108, you know that 8.076 3 107 is less than both
1.034 3 108 and 7.8 3 108. Because 1.034 < 7.8, you know that
1.034 3 108 is less than 7.8 3 108.
So, 8.076 3 107 < 1.034 3 108 < 7.8 3 108.
STEP 3 Write the original numbers in order from least to greatest.
80,760,000; 103,400,000; 7.8 3 108
EXAMPLE 4
Compute with numbers in scientific notation
Evaluate the expression. Write your answer in scientific notation.
a. (8.5 3 102)(1.7 3 106 )
AVOID ERRORS
8
Notice that 14.45 3 10
is not written in
scientific notation
because 14.45 > 10.
5 (8.5 p 1.7) 3 (102 p 106)
Commutative property and
associative property
5 14.45 3 108
Product of powers property
1
5 (1.445 3 10 ) 3 10
8
Write 14.45 in scientific notation.
5 1.445 3 (101 3 108)
5 1.445 3 10
Associative property
9
Product of powers property
b. (1.5 3 1023)2 5 1.52 3 (1023)2
Power of a product property
26
5 2.25 3 10
REVIEW FRACTIONS
For help with fractions,
see p. 915.
Power of a power property
1.2
104
1.2 3 104
c. }
5}
3}
23
23
1.6
1.6 3 10
Product rule for fractions
10
5 0.75 3 107
21
Quotient of powers property
5 (7.5 3 10 ) 3 10
7
5 7.5 3 (1021 3 107)
5 7.5 3 10
✓
GUIDED PRACTICE
6
Write 0.75 in scientific notation.
Associative property
Product of powers property
for Examples 3 and 4
2. Order 2.7 × 105, 3.401 × 104, and 27,500 from least to greatest.
Evaluate the expression. Write your answer in scientific notation.
3. (1.3 3 1025)2
4.5 3 105
4. }
22
1.5 3 10
5. (1.1 3 107)(4.2 3 102)
8.4 Use Scientific Notation
513
EXAMPLE 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 r1 be the radius of a venule, and let r 2 be the radius of a capillary.
Find the ratio of r1 to r 2 . What does the ratio tell you?
b. Let A1 be the cross-sectional area of a venule, and let A 2 be the
cross-sectional area of a capillary. Find the ratio of A1 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 r1
is 1.0 3 1022 millimeter and the radius of the capillary r 2 is
5.0 3 1023 millimeter.
r1
r2
1.0 3 1022
1.0
1022
5}
3}
5 0.2 3 101 5 2
5.0
5.0 3 10
1023
}5 }
23
The ratio tells you that the radius of the venule is twice the radius
of the capillary.
ANOTHER WAY
You can also find the
ratio of the crosssectional areas by
finding the areas using
the values for r1 and r 2,
setting up a ratio, and
then simplifying.
b. To find the cross-sectional areas, use the formula for the area of a circle.
A1
πr 12
A2
πr 2
} 5 }2
r
Write ratio.
2
1
5}
2
r2
r 2
1 2
Divide numerator and denominator by p.
1
5 }
r
Power of a quotient property
5 22 5 4
Substitute and simplify.
2
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
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
SKILL PRACTICE
1. VOCABULARY Is 0.5 3 106 written in scientific notation? Explain why or
why not.
2.
★
WRITING Is 7.89 3 106 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
12. 0.00000526
15.
★
10. 0.000045
11. 1,060,000,000
13. 900,000,000,000,000
14. 0.00000007008
MULTIPLE CHOICE Which number represents 54,004,000,000 written in
scientific notation?
A 54004 3 106
B 54.004 3 109
C 5.4004 3 1010
D 0.54004 3 1011
EXAMPLE 2
WRITING IN STANDARD FORM Write the number in standard form.
on p. 512
for Exs. 16–28
16. 2.6 3 103
17. 7.5 3 107
18. 1.11 3 102
19. 3.03 3 104
20. 4.709 3 106
21. 1.544 3 1010
22. 6.1 3 1023
23. 4.4 3 10210
24. 2.23 3 1026
25. 8.52 3 1028
26. 6.4111 3 10210
27. 1.2034 3 1026
28. ERROR ANALYSIS Describe and correct the
error in writing 1.24 3 1023 in standard form.
1.24 3 1023 5 1240
EXAMPLE 3
ORDERING NUMBERS Order the numbers from least to greatest.
on p. 513
for Exs. 29–32
29. 45,000; 6.7 3 103 ; 12,439; 2 3 104
30. 65,000,000; 6.2 3 106 ; 3.557 3 107; 55,004,000; 6.07 3 106
31. 0.0005; 9.8 3 1026 ; 5 3 1023 ; 0.00008; 0.04065; 8.2 3 1023
32. 0.0000395; 0.00010068; 2.4 3 1025 ; 5.08 3 1026 ; 0.000005
COMPARING NUMBERS Copy and complete the statement using <, >, or 5.
33. 5.6 3 103
? 56,000
35. 9.86 3 1023
37. 2.203 3 1024
? 0.00986
? 0.0000203
34. 404,000.1
? 4.04001 3 105
36. 0.003309
? 3.309 3 1023
38. 604,589,000
? 6.04589 3 107
8.4 Use Scientific Notation
515
EXAMPLE 4
EVALUATING EXPRESSIONS Evaluate the expression. Write your answer in
on p. 513
for Exs. 39–48
scientific notation.
39. (4.4 3 103)(1.5 3 1027)
40. (7.3 3 1025)(5.8 3 102)
41. (8.1 3 1024)(9 3 1026 )
6 3 1023
42. }
26
5.4 3 1025
43. }
22
44. }8
45. (5 3 1028 ) 3
46. (7 3 1025)4
47. (1.4 3 103)2
8 3 10
48.
★
1.8 3 10
1.235 3 104
MULTIPLE CHOICE Which number is the value of }
?
9.5 3 107
A 0.13 3 1024
49.
4.1 3 104
8.2 3 10
B 1.3 3 1024
C 1.3 3 1023
D 0.13 3 103
★ OPEN – ENDED
Write two numbers in scientific notation whose
product is 2.8 3 104. Write two numbers in scientific notation whose
quotient is 2.8 3 104.
50. CHALLENGE Add the numbers 3.6 3 105 and 6.7 3 104 without writing
the numbers in standard form. Write your answer in scientific notation.
Describe the steps you take.
PROBLEM SOLVING
EXAMPLE 3
51. INSECT LENGTHS The lengths of
several insects are shown in the table.
on p. 513
for Exs. 51–52
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 1021
Walking stick
555
Parasitic wasp
1.4 3 1024
Elephant beetle
1.67 3 102
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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 3 109 miles. Which spacecraft
had traveled the greater distance at that time?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
EXAMPLE 4
53. AGRICULTURE In 2002, about 9.7 3 108 pounds of cotton
were produced in California. The cotton was planted on
6.9 3 105 acres of land. What was the average number of
pounds of cotton produced per acre? Round your answer
to the nearest whole number.
on p. 513
for Ex. 53
EXAMPLE 5
on p. 514
for Exs. 54–55
516
54.
★
SHORT RESPONSE The average flow rate of the Amazon
River is about 7.6 3 106 cubic feet per second. The average
flow rate of the Mississippi River is about 5.53 3 105 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?
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.
&BSUI
.PPO
R
RKM
R
RKM
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 3 107 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 1023 inch. A print of the image measures 1 3 103 pixels by
1.5 3 103 pixels. What are the dimensions of the print in inches?
MULTIPLE REPRESENTATIONS The speed of light is
58.
1.863 3 105 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 1017 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 1022 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 3 105 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 1026 watt per square meter.
Class
Bn
2
Peak intensity (w/m )
Cn
27
n 3 10
Mn
26
Xn
25
n 3 10
n 3 10
n 3 1024
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 1011 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%
1
66. }
%
4
5
67. }
%
2
68. 133%
Graph the equation.
69. x 5 25 (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 5 22x 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 7y22)23
(6x)22y 5
1
3. }
23
4. }
3 27
7. 8.007 3 1025
8. 9.253 3 1027
Dinosaur
Mass (kilograms)
(5z)
2x y
Write the number in standard form. (p. 512)
5. 6.02 3 106
6. 5.41 3 1011
9. DINOSAURS The estimated masses of
several dinosaurs are shown in the table.
(p. 512)
Brachiosaurus
a. List the masses of the dinosaurs in
Diplodocus
1.06 3 104
Apatosaurus
29,900
Ultrasaurus
1.36 3 105
order from least to greatest.
b. Which dinosaurs are more massive
than Brachiosaurus?
518
EXTRA PRACTICE for Lesson 8.4, p. 945
77,100
ONLINE QUIZ at classzone.com
Graphing
p
g
Calculator
ACTIVITY
ACTIVITY
Use after Lesson 8.4
classzone.com
Keystrokes
8.4 Use Scientific Notation
QUESTION
How can you use a graphing calculator to solve problems
that involve numbers in scientific notation?
EXAMPLE
Use numbers in scientific notation
Gold is one of many trace elements dissolved in seawater. There is about
1.1 3 1028 gram of gold per kilogram of seawater. The mass of the oceans is
about 1.4 3 1021 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
p
Amount of
seawater in oceans
(kilograms)
(gram/kilogram)
STEP 2 Find product The product is (1.1 3 1028) p (1.4 3 1021).
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 1013.
There are about 1.54 3 1013 grams of gold
present in the oceans.
(1.1*10^-8)(1.4*10
^21)
1.54E13
PRACTICE
Evaluate the expression. Write the result in scientific notation.
1. (1.5 3 104)(1.8 3 109)
2. (2.6 3 10214)(1.4 3 1020 )
3. (7.0 3 1025) 4 (2.8 3 106 )
4. (4.5 3 1015) 4 (9.0 3 1022)
5. GASOLINE A scientist estimates that it takes about 4.45 3 107 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 1011 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 1022 atoms of carbon in 1 gram of carbon. How
many atoms of carbon were used?
8.4 Use Scientific Notation
519