1.4
Positive Integer Exponents and
Scientific Notation
1.4
OBJECTIVES
1. Use the properties of exponents
2. Use scientific notation
Exponents are used as a short-hand form for repeated multiplication. Instead of writing
aaaaa
we write
NOTE We call a the base of the
a5
expression and 5 the exponent
or the power.
which we read as “a to the fifth power.”
Definitions:
Exponential Form
In general, for any real number a and any natural number n,
an a a a
n factors
An expression of this type is said to be in exponential form. We call a the base of the
expression and n the exponent, or the power.
Let’s consider what happens when we multiply two expressions in exponential form
with the same base.
a4 a5 (a a a a)(a a a a a)
NOTE We expand the
expressions and apply the
associative property to regroup.
4 factors
5 factors
aaaaaaaaa
9 factors
a
9
Rules and Properties:
First Property of Exponents
For any real number a and natural numbers m and n,
a m a n a mn
am an (a a a)(a a a)
NOTE This is our first property
of exponents
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Notice that the product is simply the base taken to the power that is the sum of the two original exponents.
In fact, in general, the following holds:
m factors
n factors
a a a
m n factors
amn
25
26
CHAPTER 1
THE REAL NUMBERS
Example 1
Simplifying Expressions
Simplify each expression.
(a) b4 b6 b10
(b) (2a)3 (2a)4 (2a)7
(c) (2)5(2)4 (2)9
(d) 107 1011 1018
CHECK YOURSELF 1
Simplify each product.
(a) (5b)6(5b)5
(b) (3)4(3)3
(c) 108 1012
(d) (xy)2(xy)3
Applying the commutative and associative properties of multiplication, we know that a
product such as
2x3 3x2
can be rewritten as
(2 3)(x3 x2)
or as
6x5
We expand on the ideas illustrated above in our next example.
Example 2
Simplifying Expressions
Using the first property of exponents together with the commutative and associative properties, simplify each product.
and add the exponents by
Property 1. With practice you
will not need to write the
regrouping step.
(a) (5x4)(3x2) (5 3)x4x2 15x6
(b) (x2y3)(x2y4) (x2 x2)(y3 y4) x4y7
(c) (4c5d 3)(3c2d 2) (4 3)(c5c2)(d 3d 2) 12c7d 5
CHECK YOURSELF 2
Simplify each expression.
(a) (4a2b)(2a3b4)
(b) (3x4)(2x3y)
We now consider a second property of exponents that can be used to simplify quotients
of expressions in exponential form that have the same base.
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NOTE Multiply the coefficients
POSITIVE INTEGER EXPONENTS AND SCIENTIFIC NOTATION
SECTION 1.4
Consider the quotient
a6
a4
If we write this in expanded form, we have
6 factors
aaaaaa
aaaa
4 factors
This can be reduced to
NOTE Divide the numerator
and denominator by the four
common factors of a.
a
NOTE Notice that 1, when
a
a 0.
a a a a aa
a a a a
or
a2
This means that
a6
a2
a4
Rules and Properties:
Second Property of Exponents
In general, for any real number a (a 0) and natural numbers m and n, m n,
am
amn
an
NOTE This is our second
property of exponents. We
write a 0 to avoid division
by zero.
Example 3
Simplifying Expressions
Simplify each expression.
NOTE Subtract the exponents,
(a)
x10
x4
(b)
a8
a7
(d)
32a4b5
8a2b
(e)
1016
106
(a)
x10
x104 x6
x4
(b)
a8
a87 a
a7
(c)
63w8
9w85 9w3
7w5
(d)
32a4b5
4a42b51 4a2b4
8a2b
(e)
1016
10166 1010
106
© 2001 McGraw-Hill Companies
applying the second property.
NOTE Notice that a1 a; there
is no need to write the
exponent of 1 because it is
understood.
NOTE We divide the
coefficients and subtract the
exponents.
NOTE Divide the coefficients
and subtract the exponents for
each variable.
(c)
63w8
7w5
27
28
CHAPTER 1
THE REAL NUMBERS
CHECK YOURSELF 3
Simplify each expression.
(a)
y12
y5
(b)
x9
x8
(d)
49a6b7
7ab3
(e)
1013
105
(c)
45r8
9r6
Suppose that we have an expression of the form
(a2)4
This can be written as
or
a8
(a a)(a a)(a a)(a a)
2 4 or 8 factors
This suggests in general, the following:
Rules and Properties:
Third Property of Exponents
For any real number a and natural numbers m and n,
NOTE This is our third property
(am)n amn
of exponents.
Our next example illustrates the use of this third property of exponents.
Example 4
Simplifying Expressions
Using Property 3 of exponents, simplify each expression.
(a) (x3)5 x35 x15
(b) (a2)8 a28 a16
(c) (102)3 1023 106
Be Careful! Students sometimes confuse (x3)5 or x15 with x3 x5 or x8. In the first case we
multiply the exponents, in the second we add!
CHECK YOURSELF 4
Simplify each expression.
(a) (b4)7
(b) (103)3
(c) b4b7
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NOTE We multiply the
exponents.
POSITIVE INTEGER EXPONENTS AND SCIENTIFIC NOTATION
SECTION 1.4
29
Let’s develop another property for exponents. An expression such as
(2x)5
can be written in expanded form as
(2x)(2x)(2x)(2x)(2x)
5 factors
We could use the commutative and associative properties to write this product as
(2 2 2 2 2)(x x x x x)
or
NOTE Notice that each factor
25x5
of the base has been raised to
the fifth power.
This suggests our fourth property of exponents.
Rules and Properties:
Fourth Property of Exponents
For any real numbers a and b and any natural number m,
NOTE This is our fourth
(ab)m ambm
property of exponents.
The use of this fourth property is illustrated in our next example.
Example 5
Simplifying Expressions
Simplify each expression.
(a) (xy)5 x5y5
(b) (10a)4 104 a4 10,000a4
NOTE Notice that we also
apply the third property in
simplifying this expression.
(c) (2p2q3)3 23( p2)3(q3)3
8p6q9
© 2001 McGraw-Hill Companies
CHECK YOURSELF 5
Simplify each expression.
(a) (ab)7
(b) (4p)3
(c) (3m4n2)2
Our fifth (and final) property of exponents can be established in a similar fashion to the
fourth property. It deals with the power of a quotient rather than the power of a product.
30
CHAPTER 1
THE REAL NUMBERS
Rules and Properties:
Fifth Property of Exponents
For any real numbers a and b (b 0) and natural number m,
b
a
NOTE This is our fifth property
of exponents.
m
am
bm
Our next example shows the application of this property.
Example 6
Simplifying Expressions
Simplify each expression.
x2
(x2)3
x6
3 3
y
y
(2a)4
(b3)4
Property 5
24a4
16a4
12 b
b12
Property 4, Property 3
3
(a)
y
(b)
b 2a
4
3
Property 5, Property 3
CHECK YOURSELF 6
Simplify each expression.
(a)
m3
n
4
(b)
3t 2
s3
3
As we have seen, more complicated expressions require the use of more than one of our
properties, for simplification. The next example illustrates other such cases.
Example 7
Simplifying Expressions
Use the properties of exponents to simplify the following expressions.
2 x 64x
6 6
6
Property 1
Property 4
(b)
(x4)3
x12
x126 x6
3 2 (x )
x6
Property 3, Property 2
(c)
6a4b5
2a42b51 2a2b4
3a2b
Property 2, Division
(d)
7.5 1014
7.5
10143
3 2.5 10
2.5
Property 2
3 1011
Divide.
© 2001 McGraw-Hill Companies
(a) (2x)2(2x)4 (2x)24 (2x)6
POSITIVE INTEGER EXPONENTS AND SCIENTIFIC NOTATION
SECTION 1.4
31
CHECK YOURSELF 7
Simplify each expression.
(a) (3y)2(3y)3
(b)
(3a 2)3
9a3
(c)
25x3y4
5x2y
The following table summarizes the five properties of exponents introduced in this
section.
General Form
Example
I. a a a
x2 x3 x5
m n
II.
mn
am
am n a 0, m n
an
57
54
53
III. (am)n amn
(z5)4 z20
IV. (ab)m ambm
(4x)3 43x3 64x3
V.
a
b
m
am
bm
b0
2
3
6
26
64
6 3
729
Before leaving the properties of exponents, we would like to make an important extension
of one of the properties. In Property 2, suppose that we now allow m to equal n. We then
have
am
amm a0
am
(1)
But we know that it is also true that
am
1
am
(2)
Comparing (1) and (2), it then seems reasonable to make the following definition.
NOTE With this definition
Definitions: The Zero Exponent
am
amn when m n.
an
For any real number a, a 0,
© 2001 McGraw-Hill Companies
NOTE We must have a 0
a0 1
because the form 00 is called
indeterminate. It is considered
in later mathematics classes.
Example 8
Using Zero as an Exponent
Use the above definition to simplify each expression.
0
NOTE Notice that in 6x the
zero exponent applies only to x.
(a) 100 1
(c) 6x0 6 1 6
(b) (a3b2)0 1
32
CHAPTER 1
THE REAL NUMBERS
CHECK YOURSELF 8
Simplify each expression.
(a) 250
(b) (m4n2)0
(c) 8s0
You may have noticed that throughout this section we have frequently used 10 as a base
in our examples. You will find that experience useful as we discuss scientific notation.
We begin the discussion with a calculator exercise. On most (scientific) calculators, if
you find 2.3 times 1000, the display will read
2300.
Multiply by 1000 a second time. Now you will see
2300000.
Multiplying by 1000 a third time will result in the display
NOTE This must equal
2.3 09
2,300,000,000.
And multiplying by 1000 again yields
2.3 12
table
2.3 2.3 100
23 2.3 101
230 2.3 102
2300 2.3 103
23,000 2.3 104
230,000 2.3 105
Can you see what is happening? This is the way calculators display very large numbers.
The number on the left is always between 1 and 10, and the number on the right indicates
the number of places the decimal point must be moved to the right to put the answer in standard (or decimal) form.
This notation is used frequently in science. It is not uncommon, in scientific applications
of algebra, to find yourself working with very large or very small numbers. Even in the time
of Archimedes (287–212 B.C.), the study of such numbers was not unusual. Archimedes
estimated that the universe was 23,000,000,000,000,000 meters in diameter.
In scientific notation, his estimate for the diameter of the universe would be
2.3 1016 m
In general, we can define scientific notation as follows:
Definitions: Scientific Notation
Any number written in the form
a 10n
in which 1 a 10 and n is an integer, is written in scientific notation.
Example 9
Writing Numbers in Scientific Notation
Write each of the following numbers in scientific notation.
(a) 120,000
(b) 88,000,000
(c) 520,000,000
(d) 4,000,000,000
© 2001 McGraw-Hill Companies
NOTE Consider the following
POSITIVE INTEGER EXPONENTS AND SCIENTIFIC NOTATION
SECTION 1.4
33
(a) 120000. 1.2 105
NOTE Notice the pattern for
writing a number in scientific
notation.
Five places
The power is 5.
NOTE The exponent on the 10
shows the number of places we
must move the decimal point so
that the multiplier will be a
number between 1 and 10.
(b) 88,000,000. 8.8 107
Seven places
The power is 7.
(c) 520,000,000. 5.2 108
Eight places
NOTE To convert back to
standard or decimal form, the
process is simply reversed.
(d) 4,000,000,000. 4 109
Nine places
CHECK YOURSELF 9
Write in scientific notation.
(a) 212,000,000,000,000,000
(b) 5,600,000
Example 10
Applying Scientific Notation
Light travels at a speed of 3.05 108 meters per second (m/s). There are approximately
3.15 107 s in a year. How far does light travel in a year?
We multiply the distance traveled in 1 s by the number of seconds in a year. This yields
NOTE Multiply the coefficients,
(3.05 108)(3.15 107) (3.05 3.15)(108 107)
add the exponents.
9.6075 1015
NOTE Notice that
9.6075 10 10 10 10
15
15
16
For our purposes we round the distance light travels in a year to 1016 m. This unit is called
a light-year, and it is used to measure astronomical distances.
© 2001 McGraw-Hill Companies
Example 11
Applying Scientific Notation
The distance from Earth to the star Spica (in Virgo) is 2.2 1018 m. How many light-years
is the star Spica away from Earth?
NOTE We divide the distance
(in meters) by the number of
meters in 1 light-year.
2.2 1018
2.2 101816
1016
2.2 102 220 light-years
CHAPTER 1
THE REAL NUMBERS
CHECK YOURSELF 10
The farthest object that can be seen with the unaided eye is the Andromeda galaxy.
This galaxy is 2.3 1022 m from Earth. What is this distance in light-years?
CHECK YOURSELF ANSWERS
1. (a) ( 5b)11; (b) (3)7; (c) 1020; (d) (xy)5
2. (a) 8a5b5; (b) 6x7y
7
2
5 4
8
3. (a) y ; (b) x; (c) 5r ; (d) 7a b ; (e) 10
4. (a) b28; (b) 109; (c) b11
m12
27t6
5. (a) a7b7; (b) 64p3; (c) 9m8n4
6. (a) 4 ; (b) 9
n
s
5
3
3
7. (a) 243y ; (b) 3a ; (c) 5xy
8. (a) 1; (b) 1; (c) 8
9. (a) 2.12 1017; (b) 5.6 106
10. 2.3 106 or 2,300,000 light-years
© 2001 McGraw-Hill Companies
34
Name
Exercises
1.4
Section
Date
Write each product in exponential form. Identify the base and the exponent.
1. a a a a
2. x x x x x x
3. 2 2 2
4. 10 10 10 10 10
5. (3x)(3x)(3x)(3x)
6. (5p)(5p)(5p)(5p)(5p)
ANSWERS
1.
2.
3.
Simplify each of the following products.
4.
7. y y
3
8. x x
4
4
9. p6 p5 p4
6
5.
10. a2 a3 a4 a5
6.
11. 25 20
12. (2)6(2)4
7.
8.
13. (3)3(3)4(3)6
14. 75 74 72 70
9.
10.
15. 2 a5 a3 a5
16. 4 b5 b2 b4
11.
12.
13.
14.
Use the first property of exponents together with the commutative and associative properties to simplify the following products.
15.
16.
17. (x2 y3)(x3y2)
18. (a5b)(a2b4)
17.
18.
19. (m3n2)(m5n3)(m2n3)
20. (p2q3)( p4q)( p5q0)
19.
20.
21. (2a5)(4a3)(3a3)
22. (2b4)(4b)(3b5)
21.
22.
23. (6s2)(3s4)(s0)(2s2)
24. (2a2)(4a)(a3)(2a2)
23.
24.
25.
26.
25. (5xy3)(3x3y)(2xy)
26. (2ab)(6a3b)(3a2b0)
27.
28.
29.
30.
31.
32.
33.
34.
2
2 4
5
27. (rs t)(r s t)(r st)
7 4 6
4
28. (xyz)(x y z )(x yz)(xyz )
Use the second property of exponents to simplify each expression.
© 2001 McGraw-Hill Companies
3
29.
a10
a6
30.
x24
x17
31.
a6b12
a4b4
32.
x6y9
xy4
33.
x6y5z2
x2yz0
34.
a8b5c3
a2bc0
35
ANSWERS
35.
36.
35.
28m2n6
7mn2
36.
36r6s6
12r2s
37.
(2y 1)8
(2y 1)6
38.
(2x 3)5
(2x 3)4
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
49.
51.
Use the properties of exponents to simplify each expression.
39. (a6)3
40. ( p6)4
41. (3x2)(x3)4
42. (b2)(2b2)3
48.
43. (2w 3)(w4)(w2)
44. (5x2)(3x5)(x)
50.
45. (m2n)(m4)2(m4)0
46. (a3)2(a4b)(b2)4
47. (rs3t)(r3)3(s3)2(t4)2
48. (a3bc2)(a3)(b2)3(c3)0
52.
53.
Simplify each expression.
54.
49. (3b4)3(b2)3
50. (2a3)2(a2)3(a4)
51. (3w2)2(2w2)3
52. (1 103)2 (2 102)4
53. (x2y4)3(xy2)0
54. (m3n4)4(m2n)3
58.
55. (ab4c)4(abc3)8(a6bc)5
56. (x3y3z3)0(xy3z)2(x4yz3)
59.
57. (3a3)(2b4)3
58. (2m3)(5m3)2
59. (3b3)2(b0)6
60. (4a0)2(a4)3
61. (2x3)4(3x4)2
62. (3a2)3(2a5)3
55.
56.
57.
60.
61.
62.
2x5
y3
2
63.
65.
n m 64.
2a5
3b8
3
x9
3y5
2
3
64.
66.
y 2x 65.
66.
36
m8
3n7
4
3
© 2001 McGraw-Hill Companies
63.
ANSWERS
67. (8m2n)(3m4n5)4
68. (5a5b)(3a3b4)3
69. w3(3w3)2(2w4)3
70. x3(4x4)(3x0)3
8 3
5 4
67.
68.
69.
71.
6 5 2
2x y a b 3a b
2 7
xy
3 0
72.
3
5cd xy 6x y
cd
3
70.
71.
Express each number in scientific notation.
73. The distance from Mars to the sun: 141,000,000 mi
72.
73.
74.
75.
76.
77.
74. The diameter of Earth’s orbit: 186,000,000 mi
75. The diameter of Jupiter: 88,000 mi
76. The amount of free oxygen on Earth: 1,500,000,000,000,000,000,000 g
78
79.
80.
77. The mass of the moon is approximately 7.37 1022 kg. If this were written in
© 2001 McGraw-Hill Companies
standard or decimal form, how many zeros would follow the second digit 7?
78. Scientists estimate the mass of our sun to be 1.98 1024 kg. If this number
were written in standard or decimal form, how many zeros would follow the digit 8?
79. The distance from Pluto to the sun is 5.91 1012 mi. If this number were written in
standard or decimal form, how many zeros would follow the digit 1?
80. The distance light travels in 100 years is 5.8 1014 mi. If this number were written in
standard or decimal form, how many zeros would follow the digit 8?
37
ANSWERS
81.
In the expressions below, perform the indicated calculations. Write your result in scientific
notation.
82.
81. (2 105)(3 103)
83.
84.
85.
86.
82. (3.3 107)(2 104)
83.
9 109
3 106
84.
7.5 1011
1.5 107
85.
(3.3 1015)(9 1010)
(1.1 108)(3 106)
86.
(6 1012)(4.8 106)
(1.6 107)(3 102)
87. Alkaid, the most distant star in the Big Dipper, is 2.1 1018 m from Earth.
Approximately how long does it take light, traveling at 1016 m/year, to travel from
Alkaid to Earth?
87.
88. Megrez, the nearest of the Big Dipper stars, is 6.6 1017 m from Earth.
88.
Approximately how long does it take light, traveling at 1016 m/year, to travel from
Megrez to Earth?
89.
90.
Do each of the following problems.
91.
89. Write 83 as a power of 2.
92.
90. Write 166 as a power of 2.
(Remember that 8 2 .)
3
93.
91. Write 912 as a power of 3.
92. Write 818 as a power of 3.
94.
93. Write 38 as a power of 9.
94. Write 318 as a power of 27.
95.
95. Write 29 as a power of 8.
96. Write 1020 as a power of 100.
Assume that n is an integer such that all exponents are positive numbers. Then simplify
each of the following expressions.
rn4
97. a2n a4n
98. x n1 x2n
99. n1
100. (w n)4n
r
96.
97.
98.
101. (an2) n
99.
102.
(x3n)(xn3)
x4n
103.
(wn)(w4n5)
w5n
104. Do some research to discover the meaning of the word “googol.” What are the
100.
origins of this term? What connection does it have with this section?
101.
102.
103.
1. a4, base a, exponent 4
3. 23, base 2, exponent 3
5. (3x)4, base 3x, exponent 4
7
15
5
13
13
7. y
9. p
11. 2
13. (3)
15. 2a
17. x5y5
10 8
11
8
5 5
19. m n
21. 24a
23. 36s
25. 30x y
27. r8s7t3
4
2 8
4 4 2
4
29. a
31. a b
33. x y z
35. 4mn
37. (2y 1)2
39. a18
14
9
10
10 9 9
18
41. 3x
43. 2w
45. m n
47. r s t
49. 27b
51. 72w10
6 12
42 29 33
3 12
6
53. x y
55. a b c
57. 24a b
59. 9b
61. 144x20
104.
4x10
3a2b3x10y3
5 3
18 21
21
65.
3m
n
67.
648m
n
69.
72w
71.
y6
2
73. 1.41 108
75. 8.8 104
77. 20
79. 10
81. 6 108
83. 3 103
85. 9 1011
87. 210 years
89. 29
91. 324
4
3
6n
3
n22n
93. 9
95. 8
97. a
99. r
101. a
103. w5
63.
38
© 2001 McGraw-Hill Companies
Answers