Write each expression using a positive
exponent.
1.
SOLUTION:
=
2.
SOLUTION:
=
3.
SOLUTION:
=
4.
SOLUTION:
=
Write each fraction as an expression using a
negative exponent other than −1.
5.
SOLUTION:
=
6.
SOLUTION:
=
7.
SOLUTION:
8.
SOLUTION:
9.BASEBALL When a baseball is hit, it comes in
contact with the bat for less than 0.001 of a second.
Write 0.001 using a negative exponent other than –1.
SOLUTION:
Evaluate each expression if x = −4 and y = 2.
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each expression using a positive
exponent.
14.
SOLUTION:
=
15.
SOLUTION:
16.
SOLUTION:
=
17.
SOLUTION:
=
18.
SOLUTION:
=
19.
SOLUTION:
=
20.
SOLUTION:
=
21.
SOLUTION:
=
Write each fraction as an expression using a
negative exponent other than −1.
22.
SOLUTION:
=
23.
SOLUTION:
=
24.
SOLUTION:
=
25.
SOLUTION:
=
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
Write each decimal using a negative exponent.
30.SPACE The minimum thickness of Saturn’s A ring
is one tenth kilometer.
SOLUTION:
31.SCIENCE The diameter of a typical atom is
0.00000001 centimeter.
SOLUTION:
Evaluate each expression if n = 3, p = −2, and q
= 6.
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.SCIENCE The table below shows the average
lengths of different objects.
a. How many times as long is a virus than an atom?
b. About how many viruses would fit across a
pinhead?
c. A football field is about 102 meters long. How
many times as long is this than a cell?
SOLUTION:
a. To find how many times longer a virus is than an
atom, divide the length of a virus by the length of an
atom.
So, a virus is 103 or 1000 times as long as an atom.
b. To find how many viruses would fit across a
pinhead, divide the length of a pinhead by the length
of a virus.
So, 104 or 10,000 viruses would fit across a pinhead.
c. To find how many times as long a football field is
than a cell, divide the length of a football field by the
length of a cell.
So, a football field is 106 or 1,000,000 times as long
as a cell.
42.SCIENCE The shortest period of time ever
measured directly was a light burst of a laser lasting
about 0.000000000000001 second. Write this decimal
as a fraction and as a power of ten.
SOLUTION:
43.PHYSICAL SCIENCE The pH of a substance is a
measure of its acidity. The pH scale ranges from 0
to 14, with a pH of 7 being neutral. As the pH
decreases, the substance is more acidic. The table
shows the pH of several common substances.
a. Which substance in the table has the greatest
hydrogen ion concentration? How many times as
great is that hydrogen ion concentration than that of
egg whites?
b. Which substance has a hydrogen ion
concentration of one millionth?
c. As the pH increases by 1, describe what happens
to the concentration of hydrogen ions.
d. How many times as great is the hydrogen ion
concentration of coffee as the hydrogen ion
concentration of pure water?
SOLUTION:
a. Since is the largest value in the hydrogen ion
concentration column, coffee has the greatest
hydrogen ion concentration.
To find how many times as great the hydrogen ion
concentration of coffee is than that of egg whites,
divide the concentration for coffee by the
concentration for egg whites.
So, the hydrogen ion concentration of coffee is 103
or 1000 times as great as that of egg whites.
b. One million = 1,000,000 or 106, so one millionth =
or 10−6. Milk has a hydrogen ion
concentration of 10−6 or one millionth.
c. Compare the hydrogen ion concentration of milk,
pH 6, to that of coffee which has a pH 5.
As the pH increases by 1, the concentration of
hydrogen ions is multiplied by ordividedby10.
d. To find how many times as great the hydrogen ion
concentration of coffee is than that of pure water,
divide the concentration for coffee by the
concentration for pure water.
So, the hydrogen ion concentration of coffee is 102
or 100 times as great as that of pure water.
44.SAND A grain of sand has a volume of about
cubicmillimeters.
a. Write this number using a negative exponent.
b. An empty bottle used to create sand art can hold
about 1010 grains of sand. What is the approximate
volume of the sand art bottle?
c. If one cubic centimeter is equal to 103 cubic
millimeters, how many cubic centimeters of sand will
the bottle hold?
SOLUTION:
a.
b. To find the approximate volume of the sand art
bottle, multiply the number of grains of sand the
bottle can hold by the volume of one grain of sand.
So, the approximate volume of the sand art bottle is
106 mm3.
c. To find how many cubic centimeters of sand the
bottle will hold, divide the volume of the bottle by the
number of cubic millimeters in one cubic centimeter.
The bottle will hold 103 or 1000 cm3 of sand.
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore negative exponents. When
using powers of 10, 10−1 = or 0.1.
a. Tabular Copy and complete the table shown.
b. Verbal Do you notice a pattern between the
negative powers of 10 and their decimal equivalents?
Explain.
c. Verbal Write a verbal rule that could be used to
find the decimal equivalent of any negative power of
10.
d. Numerical Use the rule from part c to find the
value of 10−12.
SOLUTION:
a.
b. There is a pattern. Sample answer: As the
exponents decrease, the number of zeros in the
decimal places increase.
c. Sample answer: The number of zeros in the
decimal equivalent is equal to one less than the
absolute value of the negative exponent. For
example, 10−3 = 0.001.
d. 10−12 should have or12zerosinthe
decimal equivalent. 10−12 = 0.000000000001
46.SCIENCE The wavelength of x-rays are between 1
and 10 nanometers. If a nanometer is equal to a
billionth of a meter, express the greatest wavelength
of an x-ray in meters. Write the expression using a
negative exponent.
SOLUTION:
1 nanometer = meter
To express the greatest wavelength of an X-ray in
meters, multiply the greatest wavelength in
nanometers by the number of meters in 1 nanometer.
The greatest wavelength of an X-ray is 10−8 meters.
Use the Product of Powers and Quotient of
Powers rules to simplify each expression.
47.
SOLUTION:
48.
SOLUTION:
49.
SOLUTION:
50.
SOLUTION:
51.OPEN ENDED Write a power that has a negative
exponent and show the steps you take to write the
power as a fraction.
SOLUTION:
Sample answer:
52.ERRORANALYSIS Jeannette and Mahala are
evaluating the expression 2 •4−2. Is either of them
correct? Explain your reasoning.
SOLUTION:
Mahala did not follow the order of operations. She
multiplied the whole numbers first. She should have
first performed the operation with the exponent and
then multiplied. Jeannette has the correct solution.
53.REASONING Consider the following sets of
numbers:
Set 1: 2−2, (−2)−2, (−2)2, 22
Set 2: 2−3, (−2)−3, (−2)3, 23
a. Simplify each expression in Set 1. Which
expressions, if any, are equal?
b. Simplify each expression in Set 2. Which
expressions, if any, are equal?
c. Explain why the number of equal expressions is
different for each list.
d. Finish the conjecture: 2−x = (−2)−x , if and only if
___________.
e. Finish the conjecture: (−2)x = 2x,ifandonlyif
__________.
SOLUTION:
a.
22 = 4
2−2 = (−2)−2 and (−2)2 = 22
b.
None of the expressions are equal.
c. Sample answer: When you square either a positive
or a negative value, the answer is positive. When
you cube a positive value, you get a positive and
when you cube a negative value, you get a negative.
d. 2−x = (−2)−x , if and only if x is an even number.
e. (−2)x = 2x , if and only if x is an even number.
54.CHALLENGE Compare and contrast x−n and xn
where x≠0.Thengiveanumericalexampleto
show the relationship.
SOLUTION:
They are multiplicative inverses. Sample answer: 2−4
and 24 are multiplicative inverses because 2−4 = ,
and .
55.REASONING Investigate the fraction . Does it
increase or decrease as the value of n increases?
Explain.
SOLUTION:
Sample answer: If n = 3, Ifn = 4,
So,asthevalueofn increases, the
value of decreases.
56.WRITING IN MATH Explain the difference
between the expressions (−3)4 and 3−4.
SOLUTION:
Sample answer: (−3)4 is the same as (−3)(−3)(−3)
(−3) or 81. 3−4 is the same as or .
57.DNA contains the genetic code of an organism. The
length of a DNA strand is about 10−7 meter. Which
of the following represents the length of the DNA
strand as a decimal?
A 0.00001 m
B 0.000001 m
C 0.0000001 m
D 0.00000001 m
SOLUTION:
The length of the DNA strand is 0.0000001 meters.
Choice C is correct.
58.When simplified, 2−5 is equal to which of the
following?
F −32
G −
H
J 32
SOLUTION:
Choice H is correct.
59.Which of the following shows the expressions 40,
4−2, 42, and 4−3 in order from least to greatest?
A 4−3, 4−2, 42, 40
B 40 , 4−2, 4−3, 42
C 42, 40, 4−2, 4−3
D 4−3, 4−2, 40, 42
SOLUTION:
Since the bases are all equal, arrange the exponents
from least to greatest.
−3 < −2 < 0 < 2, so 4−3 < 4−2 < 40 < 42
Choice D is correct.
60.SHORT RESPONSE Ittakeslight5.3×0.000001
seconds to travel one mile. Write 0.000001 as a
fraction and as a power of 10.
SOLUTION:
61.ARTS AND CRAFTS When a piece of paper is
cut in half, the result is two smaller pieces of paper.
When the two smaller pieces are stacked and then
cut, the result is four pieces of paper. The number of
resulting sheets of paper after c cuts is 2c.
a. How many more pieces of paper are there if a
piece of paper is cut and stacked 8 times than when
a piece of paper is cut and stacked 5 times?
b. A stack of 500 sheets of notebook paper is about
1 inch thick. How thick would your stack be if you
were able to make 10 cuts?
SOLUTION:
a. After 8 cuts, there are 28 or 256 sheets.
After 5 cuts, there are 25 or 32 sheets.
So, there are 256 −32 or 224 more pieces of paper if
8 cuts are made than when 5 cuts are made.
b.
The stack would be 1024 inches or 85 feet after 10
cuts.
Cut Number Sheets of
Paper Thickness
(inches)
0 500 1
1 1000 2
2 2000 4
3 4000 8
4 8000 16
5 16,000 32
6 32,000 64
7 64,000 128
8 128,000 256
9 256,000 512
10 512,000 1024
ALGEBRA Factor each monomial.
62.
SOLUTION:
63.
SOLUTION:
64.53fg
SOLUTION:
State the slope and y-intercept of each
equation.
65.2x + y = −3
SOLUTION:
The slope is −2 and the y-intercept is −3.
66.5x + 4y = 20
SOLUTION:
The slope is − and the y-intercept is 5.
67.y = 4
SOLUTION:
The slope is 0 and the y-intercept is 4.
68.FOOD The results of a survey about favorite
hamburger condiments are shown in the table below.
Which condiment was chosen by the most people?
Explain.
SOLUTION:
Convert each part to a percent, then compare.
Mustard: 22% = 22%
Ketchup: =40%
Relish: 0.2 = 20%
Since 40% > 22% > 20%, the group that said they
prefer ketchup is largest.
Find each product.
69.25×0.001
SOLUTION:
70.107×0.0001
SOLUTION:
71.3.8×0.01
SOLUTION:
72.18×100
SOLUTION:
73.76×1000
SOLUTION:
74.134×100,000
SOLUTION:
Write each expression using a positive
exponent.
1.
SOLUTION:
=
2.
SOLUTION:
=
3.
SOLUTION:
=
4.
SOLUTION:
=
Write each fraction as an expression using a
negative exponent other than −1.
5.
SOLUTION:
=
6.
SOLUTION:
=
7.
SOLUTION:
8.
SOLUTION:
9.BASEBALL When a baseball is hit, it comes in
contact with the bat for less than 0.001 of a second.
Write 0.001 using a negative exponent other than –1.
SOLUTION:
Evaluate each expression if x = −4 and y = 2.
10.
SOLUTION:
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
Write each expression using a positive
exponent.
14.
SOLUTION:
=
15.
SOLUTION:
16.
SOLUTION:
=
17.
SOLUTION:
=
18.
SOLUTION:
=
19.
SOLUTION:
=
20.
SOLUTION:
=
21.
SOLUTION:
=
Write each fraction as an expression using a
negative exponent other than −1.
22.
SOLUTION:
=
23.
SOLUTION:
=
24.
SOLUTION:
=
25.
SOLUTION:
=
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
Write each decimal using a negative exponent.
30.SPACE The minimum thickness of Saturn’s A ring
is one tenth kilometer.
SOLUTION:
31.SCIENCE The diameter of a typical atom is
0.00000001 centimeter.
SOLUTION:
Evaluate each expression if n = 3, p = −2, and q
= 6.
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35.
SOLUTION:
36.
SOLUTION:
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.SCIENCE The table below shows the average
lengths of different objects.
a. How many times as long is a virus than an atom?
b. About how many viruses would fit across a
pinhead?
c. A football field is about 102 meters long. How
many times as long is this than a cell?
SOLUTION:
a. To find how many times longer a virus is than an
atom, divide the length of a virus by the length of an
atom.
So, a virus is 103 or 1000 times as long as an atom.
b. To find how many viruses would fit across a
pinhead, divide the length of a pinhead by the length
of a virus.
So, 104 or 10,000 viruses would fit across a pinhead.
c. To find how many times as long a football field is
than a cell, divide the length of a football field by the
length of a cell.
So, a football field is 106 or 1,000,000 times as long
as a cell.
42.SCIENCE The shortest period of time ever
measured directly was a light burst of a laser lasting
about 0.000000000000001 second. Write this decimal
as a fraction and as a power of ten.
SOLUTION:
43.PHYSICAL SCIENCE The pH of a substance is a
measure of its acidity. The pH scale ranges from 0
to 14, with a pH of 7 being neutral. As the pH
decreases, the substance is more acidic. The table
shows the pH of several common substances.
a. Which substance in the table has the greatest
hydrogen ion concentration? How many times as
great is that hydrogen ion concentration than that of
egg whites?
b. Which substance has a hydrogen ion
concentration of one millionth?
c. As the pH increases by 1, describe what happens
to the concentration of hydrogen ions.
d. How many times as great is the hydrogen ion
concentration of coffee as the hydrogen ion
concentration of pure water?
SOLUTION:
a. Since is the largest value in the hydrogen ion
concentration column, coffee has the greatest
hydrogen ion concentration.
To find how many times as great the hydrogen ion
concentration of coffee is than that of egg whites,
divide the concentration for coffee by the
concentration for egg whites.
So, the hydrogen ion concentration of coffee is 103
or 1000 times as great as that of egg whites.
b. One million = 1,000,000 or 106, so one millionth =
or 10−6. Milk has a hydrogen ion
concentration of 10−6 or one millionth.
c. Compare the hydrogen ion concentration of milk,
pH 6, to that of coffee which has a pH 5.
As the pH increases by 1, the concentration of
hydrogen ions is multiplied by ordividedby10.
d. To find how many times as great the hydrogen ion
concentration of coffee is than that of pure water,
divide the concentration for coffee by the
concentration for pure water.
So, the hydrogen ion concentration of coffee is 102
or 100 times as great as that of pure water.
44.SAND A grain of sand has a volume of about
cubicmillimeters.
a. Write this number using a negative exponent.
b. An empty bottle used to create sand art can hold
about 1010 grains of sand. What is the approximate
volume of the sand art bottle?
c. If one cubic centimeter is equal to 103 cubic
millimeters, how many cubic centimeters of sand will
the bottle hold?
SOLUTION:
a.
b. To find the approximate volume of the sand art
bottle, multiply the number of grains of sand the
bottle can hold by the volume of one grain of sand.
So, the approximate volume of the sand art bottle is
106 mm3.
c. To find how many cubic centimeters of sand the
bottle will hold, divide the volume of the bottle by the
number of cubic millimeters in one cubic centimeter.
The bottle will hold 103 or 1000 cm3 of sand.
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore negative exponents. When
using powers of 10, 10−1 = or 0.1.
a. Tabular Copy and complete the table shown.
b. Verbal Do you notice a pattern between the
negative powers of 10 and their decimal equivalents?
Explain.
c. Verbal Write a verbal rule that could be used to
find the decimal equivalent of any negative power of
10.
d. Numerical Use the rule from part c to find the
value of 10−12.
SOLUTION:
a.
b. There is a pattern. Sample answer: As the
exponents decrease, the number of zeros in the
decimal places increase.
c. Sample answer: The number of zeros in the
decimal equivalent is equal to one less than the
absolute value of the negative exponent. For
example, 10−3 = 0.001.
d. 10−12 should have or12zerosinthe
decimal equivalent. 10−12 = 0.000000000001
46.SCIENCE The wavelength of x-rays are between 1
and 10 nanometers. If a nanometer is equal to a
billionth of a meter, express the greatest wavelength
of an x-ray in meters. Write the expression using a
negative exponent.
SOLUTION:
1 nanometer = meter
To express the greatest wavelength of an X-ray in
meters, multiply the greatest wavelength in
nanometers by the number of meters in 1 nanometer.
The greatest wavelength of an X-ray is 10−8 meters.
Use the Product of Powers and Quotient of
Powers rules to simplify each expression.
47.
SOLUTION:
48.
SOLUTION:
49.
SOLUTION:
50.
SOLUTION:
51.OPEN ENDED Write a power that has a negative
exponent and show the steps you take to write the
power as a fraction.
SOLUTION:
Sample answer:
52.ERRORANALYSIS Jeannette and Mahala are
evaluating the expression 2 •4−2. Is either of them
correct? Explain your reasoning.
SOLUTION:
Mahala did not follow the order of operations. She
multiplied the whole numbers first. She should have
first performed the operation with the exponent and
then multiplied. Jeannette has the correct solution.
53.REASONING Consider the following sets of
numbers:
Set 1: 2−2, (−2)−2, (−2)2, 22
Set 2: 2−3, (−2)−3, (−2)3, 23
a. Simplify each expression in Set 1. Which
expressions, if any, are equal?
b. Simplify each expression in Set 2. Which
expressions, if any, are equal?
c. Explain why the number of equal expressions is
different for each list.
d. Finish the conjecture: 2−x = (−2)−x , if and only if
___________.
e. Finish the conjecture: (−2)x = 2x,ifandonlyif
__________.
SOLUTION:
a.
22 = 4
2−2 = (−2)−2 and (−2)2 = 22
b.
None of the expressions are equal.
c. Sample answer: When you square either a positive
or a negative value, the answer is positive. When
you cube a positive value, you get a positive and
when you cube a negative value, you get a negative.
d. 2−x = (−2)−x , if and only if x is an even number.
e. (−2)x = 2x , if and only if x is an even number.
54.CHALLENGE Compare and contrast x−n and xn
where x≠0.Thengiveanumericalexampleto
show the relationship.
SOLUTION:
They are multiplicative inverses. Sample answer: 2−4
and 24 are multiplicative inverses because 2−4 = ,
and .
55.REASONING Investigate the fraction . Does it
increase or decrease as the value of n increases?
Explain.
SOLUTION:
Sample answer: If n = 3, Ifn = 4,
So,asthevalueofn increases, the
value of decreases.
56.WRITING IN MATH Explain the difference
between the expressions (−3)4 and 3−4.
SOLUTION:
Sample answer: (−3)4 is the same as (−3)(−3)(−3)
(−3) or 81. 3−4 is the same as or .
57.DNA contains the genetic code of an organism. The
length of a DNA strand is about 10−7 meter. Which
of the following represents the length of the DNA
strand as a decimal?
A 0.00001 m
B 0.000001 m
C 0.0000001 m
D 0.00000001 m
SOLUTION:
The length of the DNA strand is 0.0000001 meters.
Choice C is correct.
58.When simplified, 2−5 is equal to which of the
following?
F −32
G −
H
J 32
SOLUTION:
Choice H is correct.
59.Which of the following shows the expressions 40,
4−2, 42, and 4−3 in order from least to greatest?
A 4−3, 4−2, 42, 40
B 40 , 4−2, 4−3, 42
C 42, 40, 4−2, 4−3
D 4−3, 4−2, 40, 42
SOLUTION:
Since the bases are all equal, arrange the exponents
from least to greatest.
−3 < −2 < 0 < 2, so 4−3 < 4−2 < 40 < 42
Choice D is correct.
60.SHORT RESPONSE Ittakeslight5.3×0.000001
seconds to travel one mile. Write 0.000001 as a
fraction and as a power of 10.
SOLUTION:
61.ARTS AND CRAFTS When a piece of paper is
cut in half, the result is two smaller pieces of paper.
When the two smaller pieces are stacked and then
cut, the result is four pieces of paper. The number of
resulting sheets of paper after c cuts is 2c.
a. How many more pieces of paper are there if a
piece of paper is cut and stacked 8 times than when
a piece of paper is cut and stacked 5 times?
b. A stack of 500 sheets of notebook paper is about
1 inch thick. How thick would your stack be if you
were able to make 10 cuts?
SOLUTION:
a. After 8 cuts, there are 28 or 256 sheets.
After 5 cuts, there are 25 or 32 sheets.
So, there are 256 −32 or 224 more pieces of paper if
8 cuts are made than when 5 cuts are made.
b.
The stack would be 1024 inches or 85 feet after 10
cuts.
Cut Number Sheets of
Paper Thickness
(inches)
0 500 1
1 1000 2
2 2000 4
3 4000 8
4 8000 16
5 16,000 32
6 32,000 64
7 64,000 128
8 128,000 256
9 256,000 512
10 512,000 1024
ALGEBRA Factor each monomial.
62.
SOLUTION:
63.
SOLUTION:
64.53fg
SOLUTION:
State the slope and y-intercept of each
equation.
65.2x + y = −3
SOLUTION:
The slope is −2 and the y-intercept is −3.
66.5x + 4y = 20
SOLUTION:
The slope is − and the y-intercept is 5.
67.y = 4
SOLUTION:
The slope is 0 and the y-intercept is 4.
68.FOOD The results of a survey about favorite
hamburger condiments are shown in the table below.
Which condiment was chosen by the most people?
Explain.
SOLUTION:
Convert each part to a percent, then compare.
Mustard: 22% = 22%
Ketchup: =40%
Relish: 0.2 = 20%
Since 40% > 22% > 20%, the group that said they
prefer ketchup is largest.
Find each product.
69.25×0.001
SOLUTION:
70.107×0.0001
SOLUTION:
71.3.8×0.01
SOLUTION:
72.18×100
SOLUTION:
73.76×1000
SOLUTION:
74.134×100,000
SOLUTION:
eSolutionsManual-PoweredbyCogneroPage1
9-4 Negative Exponents
