Download Exponent Laws and Scientific Notation Investigation 4 Investigation 4

Investigation
4
Exponent Laws and
Scientific Notation
Lucita and Tala were discussing how to multiply 4.1 × 10 4 by 3 × 10 6.
4.1 times 3 is
12.3, and we can
use a product
rule for the
powers of 10.
We can start by
rearranging
things a little.
4
6
But the answer's not in
scientific notation. The first
number is greater than 10.
4
6
)#&m&%(m&%2)#&m(&%m&%
4
6
4
6
&%(m&%2)#&m(&%m&%
) +
2&'#(m&%&%
2&'#(m&%
) +
2&'#(m&%&%
2&'#(m&%
Rewrite it as
1.23 × 1011
.
Think
& Discuss
How would you divide two numbers written in scientific notation?
14
6
For example, what is (2.12 × 10 ) ÷ (5.3 × 10 )? Write your answer
in scientific notation.
Develop & Understand: A
For the following exercises, do not use your calculator unless
otherwise indicated.
1.
Real-World Link
Estimates of the number of
stars and the number of
galaxies are revised as
astronomers gather more
data on the universe.
86
Unit E
Exponents
There are about 4 × 10 11 stars in our galaxy and about 10 11 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.
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?
Math Link
2.
You may recall that 1 billion is
1,000,000,000. What is its
equivalent in scientific notation?
3.
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.
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bacterium has been replaced by a population of 4.8 × 10 bacteria.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 4.8 × 10
E. coli bacteria? Show how you found your answer.
The speed of light is about 2 × 10 5 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 2.5 × 10 13 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.
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
Tambora, Indonesia
Santorini, Greece
Krakatoa, Indonesia
Date
Energy Released
( joules)
April 1815
about 1470 B.C.
August 1883
8 × 10 19
3 × 10 19
6 × 10 18
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
5.
A scientist is growing a culture of cells. The culture currently
12
contains 2 × 10 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 2 × 10 9 of the 2 × 10 12 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.
6..
4 × 10 28
×
-2 × 10 12
?
4 × 10 5
7.
÷
-4 × 10 12
-20
6 × 10
-10
8 × 10
8 × 10 a
8 × 10 a
Share
2 × 10 6
?
2 × 10 5
& 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 × 105) • 2 = 2 • 2 × 105 = 4 × 105
2. (6 × 10-5) • (2 × 10-7) = 6 • 2 × 10-5 • 10-7 = 1.2 × 10-12
3. (3 × 1012) - (3 × 1010) = 3 × 1012-10 = 3 × 102
88
Unit E
Exponents
Inquiry
Inquiry
Investigation
Materials
•
•
masking tape
ruler
5
Model the Solar System
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.
Average Distance
from Sun (miles)
Planet
Real-World Link
un
Ne
pt
us
an
Ur
tu
r
Sa
te
pi
Ju
s
ar
M
us
rth
Ea
M
Ve
n
er
cu
n
Su
1.
rn
e
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.
ry
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.
3.6 × 10 7
6.7 × 10 7
9.3 × 10 7
8
1.4 × 10
4.8 × 10 8
8.9 × 10 8
1.8 × 10 9
2.8 × 10 9
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
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
Go on
questions to determine the locations of the planets
in your model.
Investigation 5
89
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,
2.8 × 10 7 is written 2.8 e7.
Planet
Average Distance
from Sun
(miles)
Mercury
3.6 × 10 7
Venus
6.7 × 10 7
Earth
9.3 × 10 7
Mars
1.4 × 10 8
Jupiter
4.8 × 10 8
Saturn
8.9 × 10 8
Uranus
1.8 × 10 9
Neptune
2.8 × 10 9
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.
90
Unit E
Exponents
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?
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?
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?
5 × 107
a.
0
1 × 101
1 × 102
1 × 103
1 × 104
1 × 105
1 × 106
1 × 107
1 × 108
1 × 106
10 7
1 × 107 2 × 107 3 × 107 4 × 107 5 × 107 6 × 107 7 × 107 8 × 107 9 × 107 1 × 108
6
×
0
1
×
2 10 7
×
3 10 7
×
4 10 7
×
10 7
5
×
10 7
b.
c.
0
7 × 107
8 × 107
9 × 107
1 × 108
Investigation 5
91
On Your Own Exercises
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 1950.
2.56 × 10
9
725,000,000
Which number do you think is the world population in 1750?
In 1950? Explain your reasoning.
Real-World Link
In 2006, the United States
Census Bureau estimated the
population of New York City
6
as 8.2 × 10 residents. If each
resident produces four pounds
of trash per day, that is about
3.28 × 10 7 pounds of garbage
every year.
92
Unit E
Exponents
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?
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 = 65,536
5.
4 n is less than 1,000,000 (that is, 4 n < 1,000,000)
6.
n 2 is negative (that is, n 2 < 0)
7.
0.9 n is greater than or equal to 0 and, at the same time, 0.9 n is less
n
than or equal to 1. That is, 0 ≤ 0.9 ≤ 1.
8.
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?
On Your Own Exercises
14. Challenge
In Investigation 1, you explored positive integer powers
of 2 and of 4.
n
2
n
4
n
1
2
4
2
4
16
3
8
64
4
16
256
5
32
1,024
6
64
4,096
7
128
16,384
8
9
256
512
65,536 262,144
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 8.
-20
15.
For what positive values of x will x
16.
For what positive values of x will x
17.
For what values, positive or negative, of x will x
18.
The sixth power of 2 is 64, or 2 6 = 64.
-18
be greater than x
be greater than x
-18
-18
?
-20
?
be equal to x
-20
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.
m2
(_m1 ) 2
m
20.
x
_1 x
3
_1
3
3
()
-2
()
-x
(_m1 )
1
_
3x
-2
1
_
m
3
1 ÷ m2
2
-x
c
1 ÷ 3x
c
a
a
Prove that the second quotient law, _c = _ , works for
b
b
positive integer exponents c. Assume b is not equal to 0.
21. Prove It!
()
On Your Own Exercises
93
On Your Own Exercises
Prove that the power of a power law, (a b) c = a bc, works
for positive integer exponents b and c.
22. Challenge
Rewrite each expression using a single base and a single exponent.
23.
26.
29.
27 · 2
(-3)
3
-5
-4
81
· 2x
· (-3)
141
· 85
24.
(-4 m) 6
27.
8
55
_
25.
m 7 · 28 7
28.
( )
84 x
-
30.
9
-8
m
_a
na ÷ n 3
4a 4 · 3a 3
35.
(4x
38.
(x
-2 6
)
-2 3
) ·x
5
-3
· m4 · b7
33.
m
36.
(-m 2n 3) 4
39.
12b 5
_
4b
12
0
31.
(22 2 · 22 5)
34.
15
10n
_
37.
(a m) n · (b 3)
40.
(x y )
_
Simplify each expression.
32.
m
_
-
5n 5
2
4 -5 -3
-2
(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.
41.
×
2 10
2
-x
-2 x
?
-2 -3
42.
÷
4
-2
4x
-4 x
n7
-4 7
-x
2a
?
(2n) a
4a
47
2 2a
The speed of light is about 2 × 10 5 miles per
second. At approximately 5 × 10 13 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?
43. Physical Science
Real-World Link
Sirius, also called the Dog Star,
is a double star orbited by a
smaller star called Sirius B,
or the Pup.
94
Unit E
Exponents
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
9
1 billion. By 2000, the population was close to 6 × 10 .
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?
On Your Own Exercises
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.
÷
?
-20
3 × 10
6 × 10
3 × 10 x
?
-29
3 × 10 134
6 × 10 14
-x-1
5 × 10 a
?
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
21
with powers of various numbers. For example, 2 is 2,097,152 and
22
22
2 is 4,194,304. Using powers of 2, the number 2 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 16
48. 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 261.63 vibrations per second. The next higher C has a
frequency of about 523.25 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
On Your Own Exercises
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
$0.01
$0.02
$0.04
$0.01
$0.01 + 0.02 = $0.03
$0.03 + 0.04 = $0.07
1
2
3
4
50.
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.
Tessa: p = _r
64
2
•
• Marla: p = 64 · 2
• Antonia: p = 64 · _21
-r
r
96
Unit E
Exponents
1
Peter: p = 64 · _
2
()
r
•
• Damon: p = 64 · 0.5
• Tamera: p = 64 · (-2)
r
r
On Your Own Exercises
51.
This list of numbers continues in the same pattern in both directions.
, 1, 5, 25, 125, 625, ...
... , _
5
1
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.
b.
Using Ivana’s plan, write an expression that will give any number
on the list.
What value for n gives you 625? 1? _
?
5
1
Without computing the value of each pair of numbers, determine
which number is greater. For each exercise, explain why.
-1,600
-500
52.
2 80 or 4 42
55.
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
2
2
2
rival cake is (45 + 2) cm and is therefore only 4 cm larger than
the one he sells.
53.
3
or 27
54.
12 20 or 4 45
What computational mistake did the first baker make? What is the
actual difference in areas?
Earth travels around the sun approximately 6 × 10 8
miles each year. At approximately what speed must Earth travel in
miles per second? Give your answer in scientific notation.
56. Astronomy
97
On Your Own Exercises
57. Life Science
8 × 10
-5
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