4-4A
Learn
to operate with
scientific notation in
real-world situations.
Operating with
Scientific Notation
Masses of objects in space
are so great that astronomers
often need to use scientific
notation to describe them.
The approximate masses of
planets in the solar system
are given in the table.
NASA
Mass of Planets in the
Solar System
Planet
Approximate Mass
Mercury
3.30 1023 kg
Venus
4.87 1024 kg
Earth
5.97 1024 kg
Mars
6.42 1023 kg
Jupiter
1.89 1027 kg
Saturn
5.69 1026 kg
Uranus
8.68 1025 kg
Neptune
1.02 1026 kg
Source: www.solarsystem.nasa.gov/planets
Use the Mass of Planets table to answer questions in Example 1.
EXAMPLE
You can use the
properties of
exponents to
multiply and divide
numbers expressed in
scientific notation.
1
Division with Scientific Notation
About how many times greater is the mass of Jupiter than
the mass of Earth? Write your answer in scientific notation.
Write a unit ratio, or divide, to find how many times.
1.89 1027 kg
____________
5.97 1024 kg
1.89
1027
____
____
5.97
1024
Write the ratio of Jupiter’s mass to Earth’s mass.
Divide the coefficients and divide the powers.
0.3166 103
Subtract exponents.
2
3.166 10
Write the result in scientific notation.
Jupiter’s mass is about 3.166 102 times the mass of Earth.
You can also use a calculator to operate with
numbers in scientific notation. Recall that on
a graphing calculator, 1.89 1027 is displayed
as 1.89E27.
4-4A Operating with Scientific Notation
CC3
EXAMPLE
2
Multiplication with Scientific Notation
The mass of the Sun is approximately 3.1 106 times greater than
the mass of Mars. Find the mass of the Sun. Write your answer in
scientific notation.
mass of Sun
___________
3.1 106
6.42 1023 kg
mass of Sun
(3.1 106)(6.42 1023 kg)
3.1 6.42 10623 kg
19.902 1029 kg
1.9902 1030 kg
The ratio of the Sun’s mass to Mars’
mass is 3.1 106.
Solve for the mass of the Sun.
To multiply powers with the same
base, add exponents.
Write the result in scientific notation.
The mass of the Sun is about 1.99 1030 kg.
To add or subtract numbers in scientific notation, their powers of 10 must
be the same.
EXAMPLE
3
Addition and Subtraction with Scientific Notation
A
The mass of the Moon is about 7.35 1022 kg. Find the
approximate combined mass of Earth and its moon. Write
your answer in scientific notation.
5.97 1024 kg → 597 1022 kg
A number written
in scientific notation
has one factor
greater than or equal
to 1 and less than
10 and the other
factor a power of 10.
22
7.35 10 kg →
22
7.35 10 kg
604.35 1022 kg
6.0435 1024 kg
Rewrite so that the powers
are the same; then add.
Write the result in
scientific notation.
The combined mass of Earth and its moon is about 6.04 1024 kg.
B
How much greater is the mass of Neptune than the mass of
Venus? Write your answer in scientific notation.
1.02 1026 kg → 102.00 1024 kg Rewrite so that the powers
4.87 1024 kg → 4.87 1024 kg are the same; then subtract.
97.13 1024 kg Write the result in
9.713 1025 kg scientific notation.
Neptune’s is about 9.71 1025 kg greater in mass than Venus.
Think and Discuss
1. Explain why numbers written as 604.35 1022 or 0.317 103
are not written in scientific notation.
CC4
Chapter 4 Exponents and Roots
4-4A Exercises
GUIDED PRACTICE
Write your answer in scientific notation.
See Example 1
1. A country has a population of 1.35 107 people. The country has an area
of 5.4 104 square miles. Find the population density of the country, in
people per square mile.
See Example 2
2. The space shuttle orbits Earth at a speed of approximatately 2.8 104
kilometers per hour. The longest shuttle mission was about 4.2 102 hours
long. Find the approximate distance that shuttle traveled during its mission.
See Example 3
3. In 2007 there were about 7.21 107 dogs kept as pets in the United States.
In the same year there were about 8.17 107 cats kept as pets in the United
States. About how many dogs and cats were kept as pets in total?
INDEPENDENT PRACTICE
Write your answer in scientific notation.
See Example 1
4. The diameter of a red blood cell is 8.4 106. The diamater of the average cell
in the human body is 1 105. How many times larger is the diameter of the
average cell than the diameter of the red blood cell?
See Example 2
5. The average mass of a grain of sand on a beach is about 1.5 105 g. There
are about 5.1 1011 grains of sand in a beach volleyball court. What is the
mass of the grains of sand in the beach volleyball court?
See Example 3
6. The 2010 population of India was about 1.16 109 people. The 2010
population of the United States was about 3.1 108 people. About how
many more people lived in India than in the United States in 2010?
PRACTICE AND PROBLEM SOLVING
f
h indicated
d
d operation. Write your answer in scientific notation.
Perform
the
7. (4.84 1011) (8.8 104)
8. (2.66 1013) (9.5 105)
9. (1.44 1013) (3.5 102)
10. (8.9 107) (9.8 1010)
11. (5.01 1033) (4.1 1032)
12. (1.2 109) (7.77 1012)
13. (2.21 109) (2.6 103)
14. (2.42 1010) (5.5 104)
15. (6.3 1031) (3.5 1013)
16. (2.2 102) (4.55 1012)
17. (3.2 1017) (2.5 1016)
18. (6.52 1011) (8.08 1011)
Write your answer in scientific notation.
19. Chemistry There are about 6.022 1023 atoms of hydrogen in a mole of
hydrogen. How many hydrogen atoms are in 3.5 103 moles of hydrogen?
4-4A Operating with Scientific Notation
CC5
Write your answer in scientific notation.
21. Life Science An adult blue whale can eat 4.0 107 krill in 1 day. At that rate,
how many krill could an adult blue whale eat in 3.5 102 days?
The blue whale is
the largest animal
that has ever lived
on Earth. Weighing
up to 50 tons, it’s
as large as a jet.
22. Business A corporation has 4.8 103 employees. The average annual salary
for each of its employees is $4.5 104. Approximately how much does the
corporation pay its employees each year?
23. If there are 8.4 103 tiny insects living in each square foot of a field and the field
has 6.72 102 square feet, how many insects are living in the field?
24. A sample from the Sahara Desert contains 2.173 108 grains of sand. A sample
from the Chihuahuan Desert has 8.801 107 grains of sand. Which sample
has more grains of sand? How much more does it have?
25. Critical Thinking The average number of stars in a galaxy is about 5 1011
and the total number of galaxies is about 5 109. If you estimate the total
number of stars in the universe to be about 2.5 1021, what assumption are
you making?
26. Write About It Explain the how multiplication and division of numbers in
scientific notation is different than addition and subtraction of them.
27. What’s the Error? A student found the product of 8 106 and 5 109 to
be 4 1015. Explain the error the student made. What is the correct product?
28. Challenge Divide: (5.95 103) (1.7 107)
Test Prep and Spiral Review
29. Multiple Choice Jeffrey takes 2.4 104 steps during a long-distance run.
Each step covers an average distance of 8.1 102 mm. What total distance
did Jeffrey cover during his run?
A
2.96 101 mm
C
19.44 106 mm
B
1.944 107 mm
D
1.944 109 mm
30. Short Response The land area of Russia is approximately 1.71 107 km2.
The land area of Canada is approximately 9.99 106 km2. How much larger is
Russia than Canada?
Multiply. Write each answer in simplest form. (Lesson 2-4)
( )
11
31. 7 __
14
( )
7
32. 3 __
12
( )
33. 5 1__35
( )( )
4
34. __58 __
15
Determine whether each ordered pair is a solution of y 2x 7. (Lesson 3-1)
35. (1, 5)
CC6
36. (4, 1)
Chapter 4 Exponents and Roots
37. (11, 2)
38. (3, 13)
Denis Scott/Corbis
20. Demographics New York County in New York had a population of about
1.54 106 people in 2000. Erie County had an population of about 9.5 105
people. Find the approximate combined populations of New York and Erie
counties in 2000.
Life Science