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CHAPTER 7
The Basic Concepts of Algebra
64. Target Heart Rate To achieve the maximum benefit
from exercising, the heart rate in beats per minute
should be in the target heart rate zone (THR). For a
person aged A, the formula is
.7220 A THR .85220 A.
Find the THR to the nearest whole number for each
age. (Source: Hockey, Robert V., Physical Fitness:
The Pathway to Healthful Living, Times Mirror/
Mosby College Publishing, 1989.)
(a) 35
(b) Your age
7.5
Profit/Cost Analysis A product will produce a profit only
when the revenue R from selling the product exceeds the
cost C of producing it. In Exercises 65 and 66 find the
smallest whole number of units x that must be sold for
the business to show a profit for the item described.
65. Peripheral Visions, Inc. finds that the cost to produce
x studio quality videotapes is C 20x 100 ,
while the revenue produced from them is R 24x
(C and R in dollars).
66. Speedy Delivery finds that the cost to make x deliveries is C 3x 2300, while the revenue produced from them is R 5.50x (C and R in dollars).
Properties of Exponents and Scientific Notation
Exponents







Exponents are used to write products of repeated factors. For example, the product 3 3 3 3 is written
3 3 3 3 34.k Exponent
4 factors of 3
0.
0
..
34 3 3 3 3 81 .
The term googol, meaning 10100,
was coined by Professor Edward
Kasner of Columbia University. A
googol is made up of a 1 with one
hundred zeros following it. This
number exceeds the estimated
number of electrons in the
universe, which is 1079.
The Web search engine
Google is named after a googol.
Sergey Brin, president and
cofounder of Google, Inc., was a
math major. He chose the name
Google to describe the vast reach
of this search engine. (Source:
The Gazette, March 2, 2001.)
If a googol isn’t big enough
for you, try a googolplex:
googolplex 10
The number 4 shows that 3 appears as a factor four times. The number 4 is the exponent and 3 is the base. The quantity 34 is called an exponential expression. Read
34 as “3 to the fourth power,” or “3 to the fourth.” Multiplying out the four 3s gives
googol
.
Exponential Expression
If a is a real number and n is a natural number, then the exponential
expression an is defined as
an a a a … a .









10
1 0 000
a
Base
n factors of a
The number a is the base and n is the exponent.
EXAMPLE
1
(a) 7 7 7 49
2
Evaluate each exponential expression.
Read 72 as “7 squared.”
(b) 53 5 5 5 125 Read 53 as “5 cubed.”
(c) 24 2222 16
(d) 25 22222 32
(e) 51 5
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7.5
Properties of Exponents and Scientific Notation
361
In the exponential expression 3z 7, the base of the exponent 7 is z, not 3z.
That is,
Base is z.
3z7 3 z z z z z z z
7
while
Base is 3z.
3z 3z3z3z3z3z3z3z .
To evaluate 26, the parentheses around 2 indicate that the base is 2, so
26 222222 64 .
Base is 2.
In the expression 2 , the base is 2, not 2. The sign tells us to find the negative,
or additive inverse, of 26. It acts as a symbol for the factor 1.
6
This screen supports the results
preceding Example 2.
26 2 2 2 2 2 2 64
Base is 2.
Therefore, since 64 64, 2 2 .
6
EXAMPLE
2
6
Evaluate each exponential expression.
(a) 42 4 4 16
(b) 84 8 8 8 8 4096
(c) 24 2 2 2 2 16
There are several useful rules that simplify work with exponents. For example,
the product 25 23 can be simplified as follows.
~b k~~~~ 5 3 8 ~~~~
~~~~~b
b~~
25 23 2 2 2 2 22 2 2 28
This result—products of exponential expressions with the same base are found by
adding exponents —is generalized as the product rule for exponents.
Product Rule for Exponents
If m and n are natural numbers and a is any real number, then
am an amn.
EXAMPLE
3
Apply the product rule for exponents in each case.
(a) 34 37 347 311
(b) 53 5 53 51 531 54
(c) y 3 y 8 y 2 y 382 y 13
(d) 5y 23y 4 53y 2y 4
15y 24
15y 6
Associative and commutative properties
(e) 7p3q2p5q2 72p3p5qq 2 14p8q 3
So far we have discussed only positive exponents. How might we define a 0
exponent? Suppose we multiply 42 by 40. By the product rule,
42 40 420 42.
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CHAPTER 7
The Basic Concepts of Algebra
For the product rule to hold true, 40 must equal 1, and so we define a0 this way for
any nonzero real number a.
Zero Exponent
If a is any nonzero real number, then
a0 1.
The expression 00 is undefined.*
EXAMPLE
4
Evaluate each expression.
(a) 120 1
(c) 60 60 1
(e) 8k0 1, k 0
This screen supports the results in
parts (b), (c), and (d) of Example 4.
Base is 6.
(b) 60 1
Base is 6.
0
0
(d) 5 12 1 1 2
How should we define a negative exponent? Using the product rule again,
82 82 822 80 1.
1
is the reciprocal of 82, and a
82
number can have only one reciprocal. Therefore, it is reasonable to conclude that
1
82 2 . We can generalize and make the following definition.
8
This indicates that 82 is the reciprocal of 82. But
Negative Exponent
For any natural number n and any nonzero real number a,
an 1
.
an
With this definition, and the ones given earlier for positive and zero exponents,
the expression an is meaningful for any integer exponent n and any nonzero real
number a.
EXAMPLE
5
Write the following expressions with only positive exponents.
(a) 23 1
1
23
8
(b) 32 1
1
32
9
(c) 61 1
1
1 6
6
(d) 5z3 1
,
5z3
z0
*In advanced treatments, 00 is called an indeterminate form.
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7.5
(e) 5z3 5
1
5
3 3,
z
z
(g) m2 EXAMPLE
Properties of Exponents and Scientific Notation
(f) 5z 23 z0
1
,
5z 23
z0
1
,
m2
m0
6
Evaluate each of the following expressions.
(h) m4 (a) 31 41 1
1
4
3
7
3
4
12 12 12
(b) 51 21 1
1
2
5
3
5
2
10 10
10
31 363
1
, m0
m4
1
1
and 41 3
4
1
1
1
23
1
1
23 8
23
1
23
1
23
1
1 32 32
9
23
23
3 3
(d) 2 3
1
2 1
2
8
32
(c)
This screen supports the results in
parts (a), (b), and (d) of Example 6.
Parts (c) and (d) of Example 6 suggest the following generalizations.
Special Rules for Negative Exponents
If a 0 and b 0, then
1
an
an
and
an bm
.
bm an
A quotient, such as a8a3, can be simplified in much the same way as a product.
(In all quotients of this type, assume that the denominator is not 0.) Using the definition of an exponent,
a8 a a a a a a a a
a a a a a a 5.
a3
aaa
Notice that 8 3 5. In the same way,
a3
aaa
1
5 a5.
8 a
aaaaaaaa a
Here, 3 8 5. These examples suggest the following quotient rule for
exponents.
Quotient Rule for Exponents
If a is any nonzero real number and m and n are nonzero integers, then
am
amn.
an
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CHAPTER 7
The Basic Concepts of Algebra
EXAMPLE
7
Apply the quotient rule for exponents in each case.
Numerator exponent
b~ Denominator exponent
b
(a)
37
372
35
a
32
Minus sign
6
(b)
p
p62 p4,
p2
(d)
74
1
46
72 2
6 7
7
7
p0
(c)
1210
12109 121 12
129
(e)
k7
1
712
k 5 5,
12 k
k
k
k0
EXAMPLE
This screen supports the results in
parts (a) and (d) of Example 8.
8
Write each quotient using only positive exponents.
(a)
27
273 210
23
(b)
82
1
825 87 7
85
8
(c)
65
1
652 63 3
62
6
(d)
4
41
11
42
1 1 4
4
4
(e)
z 5
z 58 z 3,
z 8
z0
The expression 342 can be simplified as 342 34 34 344 38, where
4 2 8. This example suggests the first of the power rules for exponents; the
other two parts can be demonstrated with similar examples.
Power Rules for Exponents
If a and b are real numbers, and m and n are integers, then
amn amn,
abm ambm,
and
a
b
m
am
b 0.
bm
In the statements of rules for exponents, we always assume that zero never
appears to a negative power or to the power zero.
EXAMPLE
9
Use a power rule in each case.
2 4 24 16
4
3
3
81
(c) 3y4 34y4 81y4
(d) 6p72 62p72 62p14 36p14
2m5 3 23m5 3 23m15 8m15
(e)
, z0
z
z3
z3
z3
(a) p83 p83 p24
(b)
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7.5
Properties of Exponents and Scientific Notation
Notice that
63 1
6
3
1
and
216
2
2
3
2
3
2
365
9
.
4
These are examples of two special rules for negative exponents that apply when
working with fractions.
Special Rules for Negative Exponents
If a 0 and b 0, then
an E X A M P L E 10
and then evaluate.
(a)
2
7
3
1
a
n
a
b
and
n
b
a
n
.
Write the following expressions with only positive exponents
3
7
2
49
9
(b)
4
5
3
3
5
4
125
64
The definitions and rules of this section are summarized here.
This screen supports the results of
Example 10.
Definitions and Rules for Exponents
For all integers m and n and all real numbers a and b, the following rules
apply.
Product Rule
am an amn
Quotient Rule
am
amn
an
Zero Exponent
a0 1 (a 0)
Negative Exponent an Power Rules
Special Rules
for Negative
Exponents
(a 0)
1
(a 0)
an
amn amn
am
a m
m
b
b
abm ambm
(b 0)
1
an
bm
n
(a 0)
n a
m a
b
an
1 n
an (a 0)
a
a n
b n
(a, b 0)
b
a
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(a, b 0)
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CHAPTER 7
The Basic Concepts of Algebra
E X A M P L E 11 Simplify each expression so that no negative exponents appear
in the final result. Assume all variables represent nonzero real numbers.
(a) 32 35 325 33 1
33
1
27
or
(b) x 3 x 4 x 2 x 342 x 5 (c) 425 425 410
(e)
x 4y 2 x 4 y 2
2 5
x 2y 5
x
y
1
x5
(d) x 46 x 46 x 24 (f ) 23x 22 232 x 22
x 42 y 25
26x 4
x 6y 7
1
x 24
y7
x6
x4
26
or
x4
64
Scientific Notation Many of the numbers that occur in science are very large,
such as the number of one-celled organisms that will sustain a whale for a few hours:
400,000,000,000,000. Other numbers are very small, such as the shortest wavelength
of visible light, about .0000004 meter. Writing these numbers is simplified by using
scientific notation.
Scientific Notation
A number is written in scientific notation when it is expressed in the
form
a 10n,
where 1 a 10, and n is an integer.
As stated in the definition, scientific notation requires that the number be written as a product of a number between 1 and 10 (or 1 and 10) and some integer
power of 10. (1 and 1 are allowed as values of a, but 10 and 10 are not.) For
example, since
8000 8 1000 8 10 3,
the number 8000 is written in scientific notation as
8000 8 103.
When using scientific notation, it is customary to use instead of a dot to show
multiplication.
The steps involved in writing a number in scientific notation follow. (If the
number is negative, ignore the minus sign, go through these steps, and then attach
a minus sign to the result.)
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Properties of Exponents and Scientific Notation
367
Converting to Scientific Notation
Step 1: Position the decimal point. Place a caret, ^, to the right
of the first nonzero digit, where the decimal point will
be placed.
Step 2: Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This
number gives the absolute value of the exponent on 10.
Step 3: Determine the sign for the exponent. Decide whether
multiplying by 10 n should make the result of Step 1 larger
or smaller. The exponent should be positive to make the
result larger; it should be negative to make the result
smaller.
It is helpful to remember that for n 1, 10n 1 and 10n 10.
EXAMPLE
notation.
12
Convert each number from standard notation to scientific
(a) 820,000
Place a caret to the right of the 8 (the first nonzero digit) to mark the new location of the decimal point.
If a graphing calculator is set in
scientific notation mode, it will
give results as shown here. E5
means “times 105” and E5 means
“times 105”. Compare to the
results of Example 12.
8 20,000
^
Count from the decimal point, which is understood to be after the last 0, to the
caret.
8 20,000. k Decimal point
^
Count 5 places.
Since the number 8.2 is to be made larger, the exponent on 10 is positive.
820,000 8.2 10 5
(b) .000072
Count from left to right.
.00007 2
^
5 places
Since the number 7.2 is to be made smaller, the exponent on 10 is negative.
.000072 7.2 105
To convert a number written in scientific notation to standard notation, just work
in reverse.
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CHAPTER 7
The Basic Concepts of Algebra
Converting from Scientific Notation to Standard
Notation
Multiplying a number by a positive power of 10 makes the number
larger, so move the decimal point to the right if n is positive in 10 n.
Multiplying by a negative power of 10 makes a number smaller, so
move the decimal point to the left if n is negative.
If n is zero, leave the decimal point where it is.
EXAMPLE
13
Write each number in standard notation.
(a) 6.93 105 6.93000
5 places
The decimal point was moved 5 places to the right. (It was necessary to attach
3 zeros.)
6.93 10 5 693,000
(b) 4.7 106 000004.7
Attach 0s as necessary.
6 places
The decimal point was moved 6 places to the left. Therefore,
4.7 10 6 .0000047.
(c) 1.083 100 1.083
We can use scientific notation and the rules for exponents to simplify calculations.
1,920,000 .0015
by using scientific notation.
.000032 45,000
First, express all numbers in scientific notation.
EXAMPLE
14
Evaluate
1,920,000 .0015 1.92 106 1.5 103
.000032 45,000
3.2 105 4.5 104
1.92 1.5 106 103
3.2 4.5 105 104
1.92 1.5
104
3.2 4.5
.2 104
2 101 104
Commutative and
associative properties
Product and
quotient rules
Simplify.
2 103
2000
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7.5 Properties of Exponents and Scientific Notation
369
E X A M P L E 15 In 1990, the national health care expenditure was $695.6 billion. By 2000, this figure had risen by a factor of 1.9; that is, it almost doubled in
only 10 years. (Source: U.S. Centers for Medicare & Medicaid Services.)
(a) Write the 1990 health care expenditure using scientific notation.
695.6 billion 695.6 109 6.956 102 109
6.956 1011
Product rule
In 1990, the expenditure was $6.956 1011.
(b) What was the expenditure in 2000?
Multiply the result in part (a) by 1.9.
6.956 1011 1.9 1.9 6.956 1011
13.216 1011
Commutative and
associative properties
Round to three
decimal places.
The 2000 expenditure was $1,321,600,000,000, over $1 trillion.
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25
25
9
9
25
9
25
54
10 3
25
54
23
32
34
52
7 2
41
7 2
41
3
2
43
2
32
3
2
3
1
5
1
5
42
2
2
5
1
2
3