Download Zero and Negative Exponents and Scientific Notation

SECT
ION
10.1
10.1
10.1 O B J E C T I V E S
1. Define the zero
exponent
2. Simplify expressions with negative exponents
3. Write a number in
scientific notation
4. Solve an application of scientific
notation
Zero and Negative Exponents
and Scientific Notation
In Section 5.1, we examined properties of exponents, but all of the exponents were
positive integers. In this section, we look at zero and negative exponents. First we extend the quotient rule so that we can define an exponent of zero.
Recall that, in the quotient rule, to divide two expressions that have the same base,
we keep the base and subtract the exponents.
am
amn
an
Now, suppose that we allow m to equal n. We then have
am
m
amm a0
a
(1)
But we know that it is also true that
am
1
am
(2)
Comparing equations (1) and (2), we see that the following definition is reasonable.
The Zero Exponent
We must have a 0. The
form 00 is called
indeterminate and is
considered in later
mathematics classes.
Example 1
For any real number a where a 0,
a0 1
The Zero Exponent
Use the above definition to simplify each expression.
Note that in 6x0 the exponent
0 applied only to x.
666
(a) 170 1
(b) (a3b2)0 1
(c) 6x0 6 1 6
(d) 3y0 3
Section 10.1
■
Zero and Negative Exponents and Scientific Notation
667
✓ CHECK YOURSELF 1
■
Simplify each expression.
(a) 250
(b) (m4n2)0
(c) 8s0
(d) 7t0
Recall that, in the product rule, to multiply expressions with the same base, keep
the base and add the exponents.
am an amn
Now, what if we allow one of the exponents to be negative and apply the product rule?
Suppose, for instance, that m 3 and m 3. Then
am an a3 a3 a3(3)
a0 1
so
a3 a3 1
Negative Integer Exponents
John Wallis (1616–1702), an
English mathematician, was
the first to fully discuss the
meaning of 0, negative, and
rational exponents (which we
discuss in Section 10.3).
For any nonzero real number a and whole number n,
1
an an
and an is the multiplicative inverse of an.
Example 2 illustrates this definition.
Example 2
Using Properties of Exponents
From this point on, to simplify
will mean to write the
expression with positive
exponents only.
Simplify the following expressions.
Also, we will restrict all
variables so that they
represent nonzero real
numbers.
1
1
(b) 42 42
16
1
(a) y5 5
y
668
Chapter 10
■
Radicals and Exponents
1
1
1
(c) (3)3 3 (3)
27
27
(d)
3
2
3
1
1
27
—3 — 8
8
2
27
3
✓ CHECK YOURSELF 2
■
Simplify each of the following expressions.
(a) a10
(b) 24
(c) (4)2
(d)
2
5
2
Example 3 illustrates the case where coefficients are involved in an expression
with negative exponents. As will be clear, some caution must be used.
Using Properties of Exponents
Example 3
Simplify each of the following expressions.
1
2
(a) 2x3 2 3 3
x
x
The exponent 3 applies only
to the variable x, and
not to the coefficient 2.
!
Caution
The expressions
2
4w
and
are not the same.
Do you see why?
2
(4w)
1
4
(b) 4w2 4 2
w2
w
1
1
(c) (4w)2 2 (4w)
16w2
✓ CHECK YOURSELF 3
■
Simplify each of the following expressions.
(a) 3w4
(b) 10x5
(c) (2y)4
(d) 5t2
Section 10.1
■
Zero and Negative Exponents and Scientific Notation
669
Suppose that a variable with a negative exponent appears in the denominator of
an expression. Our previous definition can be used to write a complex fraction that can
then be simplified. For instance,
1
1
a2
2
— 1 a2
1
a
1
2
a
Negative exponent
in denominator
Positive exponent in numerator
To divide, we
invert and multiply.
To avoid the intermediate steps, we can write that, in general,
For any nonzero real number a and integer n,
1
n
an
a
Example 4
Using Properties of Exponents
Simplify each of the following expressions.
1
(a) y3
y 3
1
(b) 5
25 32
2
3
3x2
(c) 2
4x
4
The exponent 2 applies only to x, not to 4.
a3
b4
(d) 4 b
a3
✓ CHECK YOURSELF 4
■
Simplify each of the following expressions.
1
(a) x 4
To review these properties,
return to Section 5.1.
1
(b) 3
3
2
(c) 2
3a
c5
(d) d 7
The product and quotient rules for exponents apply to expressions that involve any
integral exponent—positive, negative, or 0. Example 5 illustrates this concept.
670
Chapter 10
■
Radicals and Exponents
Example 5
Using Properties of Exponents
Simplify each of the following expressions, and write the result, using positive exponents only.
(a) x3 x7 x3(7)
Add the exponents by the
product rule.
1
x4 4
x
m5
(b) m5(3) m53
m 3
Subtract the exponents by the
quotient rule.
1
m2 2
m
x5x3
x5(3)
x2
(c) x2(7) x9
x 7
x 7
x7
Note that m5 in the
numerator becomes m5 in the
denominator, and m3 in the
denominator becomes m3 in
the numerator. We then
simplify as before.
We apply first the product rule and
then the quotient rule.
In simplifying expressions involving negative exponents, there are often alternate
approaches. For instance, in Example 5, part b, we could have made use of our earlier
work to write
m5
m3
1
m35 m2 2
3 m
m5
m
✓ CHECK YOURSELF 5
■
Simplify each of the following expressions.
(a) x9 x5
y7
(b) y 3
a3a2
(c) a 5
The properties of exponents can be extended to include negative exponents. One
of these properties, the quotient-power rule, is particularly useful when rational expressions are raised to a negative power. Let’s look at the rule and apply it to negative
exponents.
Quotient-Power Rule
a
b
n
an
bn
Section 10.1
Zero and Negative Exponents and Scientific Notation
■
Raising Quotients to a Negative Power
n
a
b
Example 6
an
bn
b
n b
an
a
Extending the Properties of Exponents
Simplify each expression.
s3
2
t
2
(a)
(b)
m2
n 2
t2
3
s
3
2
t4
s6
n2
m2
3
n6
1
66
m6
mn
✓ CHECK YOURSELF 6
■
Simplify each expression.
(a)
Example 7
n
3
3t2
s3
x5
y 2
3
(b)
Using Properties of Exponents
Simplify each of the following expressions.
(a)
2
q
3
5
q5
3
2
q10
9
(b)
x3
4
y
3
y4
3
x
3
(y4)3
y12
3
3 (x )
x9
a 0, b 0
671
672
Chapter 10
■
Radicals and Exponents
✓ CHECK YOURSELF 7
■
Simplify each of the following expressions.
(a)
r4
2
5
(b)
a4
3
b
3
As you might expect, more complicated expressions require the use of more than
one of the properties, for simplification. Example 8 illustrates such cases.
Using Properties of Exponents
Example 8
Simplify each of the following expressions.
(a2)3(a3)4
a6 a12
(a) 3 3
(a )
a 9
It may help to separate the
problem into three fractions,
one for the coefficients and
one for each of the variables.
Apply the power rule to each factor.
a612
a6
a 9
a9
Apply the product rule.
a6(9) a69 a15
Apply the quotient rule.
8x2y5
8 x2 y5
(b) 4 3 12x y
12 x 4 y3
2
x2(4) y53
3
2
2x2
x2 y8 8
3
3y
(c)
pr3s5
p3r3s2
2
( p13r3(3)s5(2))2
( p2r6s3)2
( p2)2(r6)2(s3)2
Caution
!
p4s6
p4r12s6 1
r2
Apply the quotient rule inside the
parentheses.
Apply the rule for a product to a
power.
Apply the power rule.
Be Careful! Another possible first step (and generally an efficient one) is to rewrite
an expression by using our earlier definitions.
1
an n
a
and
1
n
an
a
For instance, in Example 8, part b, we would correctly write
8x2y5
8x4
4 3 2
12x y
12x y3y5
Section 10.1
■
Zero and Negative Exponents and Scientific Notation
673
A common error is to write
!
Caution
8x2y5
12x4
4 3 12x y
8x2y3y5
This is not correct.
The coefficients should not have been moved along with the factors in x. Keep in mind
that the negative exponents apply only to the variables. The coefficients remain where
they were in the original expression when the expression is rewritten using this approach.
✓ CHECK YOURSELF 8
■
Simplify each of the following expressions.
(x5)2(x2)3
(a) (x4)3
12a3b2
(b) 16a 2b3
(c)
xy3z5
3
x y z 4 2 3
Let us now take a look at an important use of exponents, scientific notation.
We begin the discussion with a calculator exercise. On most calculators, if you
multiply 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
This must equal
2,300,000,000.
2.3
09
or
2.3
E09
or
2.3
E12
And multiplying by 1000 again yields
Consider the following 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
2.3
12
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 m in di1
ameter, which is the approximate distance light travels in 2 years. By comparison,
2
Polaris (the North Star) is 680 light-years from the earth. Example 10 will discuss the
idea of light-years.
674
Chapter 10
■
Radicals and Exponents
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.
Scientific Notation
Any number written in the form
a 10n
where 1 a 10 and n is an integer, is written in scientific notation.
Example 9
Using Scientific Notation
Write each of the following numbers in scientific notation.
Note the pattern for writing a
number in scientific notation.
(a) 120,000. 1.2 105
5 places
(b) 88,000,000. 8.8 10
The power is 5.
7
7 places
The power is 7.
The exponent on 10 shows
the number of places we must
move the decimal point so
that the multiplier will be a
number between 1 and 10.
A positive exponent tells us to
move right, while a negative
exponent indicates to move
left.
(c) 520,000,000. 5.2 10
Note: To convert back to
standard or decimal form, the
process is simply reversed.
(f) 0.0000000081 8.1 109
8
8 places
(d) 4,000,000,000. 4 109
9 places
(e) 0.0005 5 x 104
4 places
If the decimal point is to be moved to the left,
the exponent will be negative.
9 places
✓ CHECK YOURSELF 9
■
Write in scientific notation.
(a) 212,000,000,000,000,000
(b) 0.00079
(c) 5,600,000
(d) 0.0000007
Section 10.1
Example 10
■
Zero and Negative Exponents and Scientific Notation
675
An Application of Scientific Notation
(a) 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
(3.05 108)(3.15 107) (3.05 3.15)(108 107)
Multiply the coefficients,
and add the exponents.
9.6075 1015
Note that 9.6075 1015 10 1015 1016
For our purposes we round the distance light travels in 1 year to 1016 m. This unit
is called a light-year, and it is used to measure astronomical distances.
(b) The distance from earth to the star Spica (in Virgo) is 2.2 1018 m. How many
light-years is Spica from earth?
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
✓ 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
1
1
4
2. (a) 10
; (b) ; (c) ; (d) .
a
16
16
25
3
10
1
5
2a2
d7
3. (a) ;
.
4. (a) x4; (b) 27; (c) ; (d) .
4 (b) ;
5 (c) ;
4 (d) 2
w
x
16y
t
3
c5
9
1
s
1
5. (a) x4; (b) ;
(c) a4.
6. (a) ;
(b) .
y4
27t6
x15y6
25
b9
3
y3z24
7. (a) ;
8. (a) x8; (b) ;
.
9. (a) 2.12 1017;
8 (b) 1.
2
5 (c) 15
r
a
4ab
x
(b) 7.9 104; (c) 5.6 106; (d) 7 107.
10. 2,300,000 light-years.
1. (a) 1; (b) 1; (c) 8; (d) 7.
E xercises
1
1. x5
1
2. 27
1
3. 25
1
4. x8
1
5. 25
1
6. 27
1
7. 8
1
8. 16
27
9. 8
16
10. 9
3
11. x2
4
12. x3
5
13. x4
1
14. 16x4
1
15. 9x2
5
16. x2
17. x3
18. x5
2x3
19. 5
3x4
20. 4
y4
21. x3
y3
22. x5
2
■
10.1
In Exercises 1 to 22, simplify each expression.
1. x5
2. 33
3. 52
4. x8
5. (5)2
6. (3)3
7. (2)3
8. (2)4
2
9. 3
3
2
3
10. 4
11. 3x2
12. 4x3
13. 5x4
14. (2x)4
15. (3x)2
16. 5x2
1
17. x 3
1
18. x 5
2
19. 3
5x
3
20. 4
4x
x3
21. y 4
x5
22. y 3
In Exercises 23 to 32, use the properties of exponents to simplify expressions.
23. x
24. y
1
25. a3
1
26. w2
23. x5 x3
24. y4 y5
25. a9 a6
26. w5 w3
1
27. 10
z
1
28. b8
27. z2 z8
28. b7 b1
29. a5 a5
30. x4 x4
29. 1
30. 1
1
31. x3
32. x3
x5
31. x 2
x3
32. x 6
676
Section 10.1
33. x15
34. w24
35. 2x5
36. 9p10
37. 3a
38. 10y4
39. x10y
40. r10s7
17 8 13
4 7 2
■
Zero and Negative Exponents and Scientific Notation
677
In Exercises 33 to 58, use the properties of exponents to simplify the following.
33. (x5)3
34. (w4)6
35. (2x3)(x2)4
36. (p4)(3p3)2
37. (3a4)(a3)(a2)
38. (5y2)(2y)(y5)
41. a b c
1
43. 15
x
42. p q r
44. x6
39. (x4y)(x2)3(y3)0
40. (r4)2(r2s)(s3)2
41. (ab2c)(a4)4(b2)3(c3)4
45. b8
1
46. 12
b
42. (p2qr2)(p2)(q3)2(r2)0
43. (x5)3
44. (x2)3
x10
47. y6
p6
48. q4
45. (b4)2
46. (a0b4)3
47. (x5y3)2
49. x12y6
27
50. 6
x y6
48. (p3q2)2
49. (x4y2)3
50. (3x2y2)3
x15
51. 32
b4
52. a6
y4
53. x2
51. (2x3y0)5
54. x9y6
a6
52. b 4
x2
53. y 4
y2
55. x4
18
56. 1
x 3
x3
54. y2
x4
55. 2
y
(3x4)2(2x2)
56. x6
48
57. x8
x
58. 20,000
59. 16x26
60. 27x16
72
61. x3
3
57. (4x2)2(3x4)
58. (5x4)4(2x3)5
In Exercises 59 to 90, simplify each expression.
8 12
62. x y
63. x42y33z25 64. x5y5z4
65. 75x2
66. 4a6
59. (2x5)4(x3)2
60. (3x2)3(x2)4(x2)
61. (2x3)3(3x3)2
62. (x2y3)4(xy3)0
63. (xy5z)4(xyz2)8(x6yz)5
64. (x2y2z2)0(xy2z)2(x3yz2)
65. (3x2)(5x2)2
66. (2a3)2(a0)5
67. (2w3)4(3w5)2
68. (3x3)2(2x4)5
3x6 y5
69. 9 3
2y x
x8 2y9
70. 6 y
x3
71. (7x2y)(3x5y6)4
2w5z3 x5y4
72. 3
3x y9 w4z0
74. (5a2b4)(2a5b0)
(x3)(y2)
75. y 3
6x3y4
76. 24x 2y2
15x3y2z4
77. 20x4y3z2
24x5y3z2
78. 36x2y3z2
x5y7
79. x0y4
2
67. 144w
68. 288x26
3x3
69. 2y4
70. 2x5y3
22 25
71. 567x y
2x7z3
72. 3w3y
6
73. 2
x y5
10a3
74. b4
y5
75. x3
x5
76. 4y2
3xy5
77. 4z6
2z4
78. 3
3x y6
1
79. 5
x y3
2
73. (2x2y3)(3x4y2)
678
Chapter 10
z12
80. x8y10
■
xy3z4
80. 3 x y2z2
y8
81. x7
82. x6y9
83. x5n
84. x4n1
1
86. x3
85. x2
2
88. xn
2
87. y3n
n
89. x2
90. x5
91. 9.3 107
92. 2.1 105 m
93. 1.3 1011
94. 6.02 10
95. 28
96. 18
97. 0.008
98. 0.0000075
99. 0.000028
100. 0.000521
4
102. 3 106
103. 3.7 104
104. 5.1 105
105. 8 10
x2y2 x4y2
81. x3y2 x2y2
2
x3y3
82. x4y2
x2y2
xy4
3
1
n3
83. x2n x3n
84. xn1 x3n
x
85. xn1
xn4
86. xn1
87. (yn)3n
88. (xn1)n
x2n xn2
89. 3
xn
xn x3n5
90. 4
xn
In Exercises 91 to 94, express each number in scientific notation.
91. The distance from the earth to the sun: 93,000,000 mi.
23
101. 5 10
Radicals and Exponents
92. The diameter of a grain of sand: 0.000021 m.
93. The diameter of the sun: 130,000,000,000 cm.
94. The number of molecules in 22.4 L of a gas: 602,000,000,000,000,000,000,000
(Avogadro’s number)
95. The mass of the sun is approximately 1.98 1030 kg. If this were written in standard or decimal form, how many 0s would follow the digit 8?
96. Archimedes estimated the universe to be 2.3 1019 millimeters (mm) in diameter. If this number were written in standard or decimal form, how many 0s would
follow the digit 3?
In Exercises 97 to 100, write each expression in standard notation.
97. 8 103
98. 7.5 106
99. 2.8 105
100. 5.21 104
8
106. 6 104
In Exercises 101 to 104, write each of the following in scientific notation.
107. 3 105
101. 0.0005
108. 5 106
In Exercises 105 to 108, compute the expressions using scientific notation, and write
your answer in that form.
109. 8 10
9
102. 0.000003
103. 0.00037
104. 0.000051
110. 7.5 1012
105. (4 103)(2 105)
106. (1.5 106)(4 102)
111. 2 102
9 103
107. 3 102
7.5 104
108. 1.5 102
112. 3 105
113. 6 1016
114. 4 10
11
In Exercises 109 to 114, perform the indicated calculations. Write your result in scientific notation.
109. (2 105)(4 104) 110. (2.5 107)(3 105)
4.5 1012
112. 1.5 107
6 109
111. 3 107
(3.3 1015)(6 1015)
(6 1012)(3.2 108)
113. 114. 8
6
(1.1 10 )(3 10 )
(1.6 107)(3 102)
Section 10.1
115. 66 years
116. 210 years
117. 1.55 1023;
2.92 1013
118. 4.66 1015 L
119. 6.5 1014
■
Zero and Negative Exponents and Scientific Notation
679
115. Megrez, the nearest of the Big Dipper stars, is 6.6 1017 m from earth. Approximately how long does it take light, traveling at 1016 m/year, to travel from
Megrez to earth?
116. Alkaid, the most distant star in the Big Dipper, is 2.1 1018 m from earth. Approximately how long does it take light to travel from Alkaid to earth?
117. The number of liters of water on earth is 15,500 followed by 19 zeros. Write this
number in scientific notation. Then use the number of liters of water on earth to
find out how much water is available for each person on earth. The population of
earth is 5.3 billion.
118. If there are 5.3 109 people on earth and there is enough freshwater to provide
each person with 8.79 105 L, how much freshwater is on earth?
119. The United States uses an average of 2.6 106 L of water per person each year.
The United States has 2.5 108 people. How many liters of water does the United
States use each year?
1
1
120. Can (a b)1 be written as by using the properties of exponents? If not,
a
b
why not? Explain.
121. Write a short description of the difference between (4)2, 43, (4)3, and
43. Are any of these equal?
122. If n 0, which of the following expressions are negative?
n3, n3, (n)3, (n)3, n3
If n 0, which of these expressions are negative? Explain what effect a negative
in the exponent has on the sign of the result when an exponential expression is
simplified.
123. Take the best offer. You are offered a 28-day job in which you have a choice of
two different pay arrangements. Plan 1 offers a flat $4,000,000 at the end of the
28th day on the job. Plan 2 offers 1¢ the first day, 2¢ the second day, 4¢ the third
day, and so on, with the amount doubling each day. Make a table to decide which
offer is the best. Write a formula for the amount you make on the nth day and a
formula for the total after n days. Which pay arrangement should you take? Why?