A. The Properties of Exponents
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The power property can easily be extended to include more than one factor within
the parentheses. This application of the power property is sometimes called the
product to a power property and can be extended to include any number of factors.
We can also raise a quotient of exponential terms to a power. The result is called the
quotient to a power property. In words the properties say, to raise a product or quotient
of exponential terms to a power, multiply every exponent inside the parentheses by the
exponent outside the parentheses.
Product to a Power Property
For any bases a and b, and positive integers m, n, and p:
1ambn 2 p ϭ amp # bnp
Quotient to a Power Property
For any bases a and b
0, and positive integers m, n, and p:
a
EXAMPLE 2
ᮣ
am p amp
b ϭ np
bn
b
Simplifying Terms Using the Power Properties
Simplify using the power property (if possible):
Ϫ5a3 2
a. 1Ϫ3a2 2
b. Ϫ3a2
c. a
b
2b
Solution
ᮣ
WORTHY OF NOTE
Regarding Examples 2a and 2b,
note the difference between the
expressions 1Ϫ3a2 2 ϭ 1Ϫ3 # a2 2 and
Ϫ3a2 ϭ Ϫ3 # a2. In the first, the
exponent acts on both the negative
3 and the a; in the second, the
exponent acts on only the a and
there is no “product to a power.”
a. 1Ϫ3a2 2 ϭ 1Ϫ32 2 # 1a1 2 2
ϭ 9a2
1Ϫ52 2 1a3 2 2
Ϫ5a3 2
b ϭ
c. a
2b
22b2
25a6
ϭ
4b2
b. Ϫ3a2 is in simplified form
Now try Exercises 13 through 24 ᮣ
Applications of exponents sometimes involve linking one exponential term with
another using a substitution. The result is then simplified using exponential properties.
EXAMPLE 3
ᮣ
Applying the Power Property after a Substitution
The formula for the volume of a cube is V ϭ S3, where S is
the length of one edge. If the length of each edge is 2x2:
a. Find a formula for volume in terms of x.
b. Find the volume if x ϭ 2.
Solution
ᮣ
a. V ϭ S
3
S
substitute 2x 2 for S
ϭ 12x2 2 3
ϭ 8x6
2x2
2x2
2x2
b. For V ϭ 8x ,
V ϭ 8122 6
substitute 2 for x
#
ϭ 8 64 or 512 122 6 ϭ 64
The volume of the cube would be 512 units3.
6
Now try Exercises 25 and 26 ᮣ
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Section R.2 Exponents, Scientific Notation, and a Review of Polynomials
13
The Quotient Property of Exponents
x
ϭ 1 for x 0, we note a
x
x5 x # x # x # x # x
ϭ x3,
pattern that helps to simplify a quotient of exponential terms. For 2 ϭ
#
x
x
x
the exponent of the final result appears to be the difference between the exponent in the
numerator and the exponent in the denominator. This seems reasonable since the subtraction would indicate a removal of the factors that reduce to 1. Regardless of how
many factors are used, we can generalize the idea and state the quotient property of
exponents. In words the property says, to divide two exponential terms with the same
base, keep the common base and subtract the exponent of the denominator from the
exponent of the numerator.
By combining exponential notation and the property
Quotient Property of Exponents
For any base b
0 and positive integers m and n:
bm
ϭ bmϪn
bn
Zero and Negative Numbers as Exponents
If the exponent of the denominator is greater than the exponent in the numerator, the
x2
quotient property yields a negative exponent: 5 ϭ x2Ϫ5 ϭ xϪ3. To help understand
x
what a negative exponent means, let’s look at the expanded form of the expression:
x2
x # x1
1
ϭ
ϭ 3 . A negative exponent can literally be interpreted as “write
5
#
#
#
#
x
x
x
x
x
x
x
the factors as a reciprocal.” A good way to remember this is
!
!
2Ϫ3
three factors of 2
written as a reciprocal
1
1
2Ϫ3
ϭ 3ϭ
1
8
2
Since the result would be similar regardless of the base used, we can generalize this
idea and state the property of negative exponents.
Property of Negative Exponents
For any base b
0 and integer n:
Ϫn
b
1
ϭ n
1
b
WORTHY OF NOTE
The use of zero as an exponent
should not strike you as strange or
odd; it’s simply a way of saying that
no factors of the base remain, since
all terms have been reduced to 1.
8
23
For 3 , we have ϭ 1, or
8
2
1
1
1
2#2#2
ϭ 1, or 23Ϫ3 ϭ 20 ϭ 1.
2#2#2
1
bϪn
ϭ
bn
1
a Ϫn
b n
a b ϭa b ;a
a
b
0
x3
x3
by
division,
and
ϭ
1
ϭ x3Ϫ3 ϭ x0 using the
x3
x3
quotient property, we conclude that x0 ϭ 1 as long as x 0. We can also generalize
this observation and state the meaning of zero as an exponent. In words the property
says, any nonzero quantity raised to an exponent of zero is equal to 1.
Finally, when we consider that
Zero Exponent Property
For any base b
0:
b0 ϭ 1
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EXAMPLE 4
Solution
ᮣ
ᮣ
Simplifying Expressions Using Exponential Properties
Simplify using exponential properties. Answer using positive exponents only.
2a3 Ϫ2
a. a 2 b
b. 13hkϪ2 2 3 16hϪ2kϪ3 2 Ϫ2
b
1Ϫ2m2n3 2 5
c. 13x2 0 ϩ 3x0 ϩ 3Ϫ2
d.
14mn2 2 3
2a3 Ϫ2
b2 2
a. a 2 b ϭ a 3 b property of negative exponents
b
2a
1b2 2 2
ϭ 2 3 2 power properties
2 1a 2
ϭ
b4
4a6
result
b. 13hkϪ2 2 3 16hϪ2kϪ3 2 Ϫ2 ϭ 33h3 1kϪ2 2 3 # 6Ϫ2 1hϪ2 2 Ϫ2 1kϪ3 2 Ϫ2
3 3 Ϫ6
ϭ3hk
Ϫ2
ϭ3
3
#6
Ϫ2 4 6
#6 hk
# h3ϩ4 # kϪ6ϩ6
ϭ
27h k
36
1
32
1
9
Notice in Example 4(c), we have
13x2 0 ϭ 13 # x2 0 ϭ 1, while
3x0 ϭ 3 # x0 ϭ 3112. This is another
example of operations and
grouping symbols working
together: 13x2 0 ϭ 1 because any
quantity to the zero power is 1.
However, for 3x0 there are no
grouping symbols, so the exponent
0 acts only on the x and not the 3:
3x0 ϭ 3 # x0 ϭ 3112 ϭ 3.
1Ϫ2m2n3 2 5
14mn2 2 3
ϭ
1Ϫ22 5 1m2 2 5 1n3 2 5
43m3 1n2 2 3
result
power property
Ϫ32m10n15
64m3n6
simplify
ϭ
Ϫ1m7n9
2
quotient property
m7n9
2
1
b
36
zero exponent property; property of negative exponents
ϭ
ϭϪ
ϭ
simplify: (3x )0 ϭ 1, 3x 0 ϭ 3 и 1 ϭ 3
1
37
ϭ4 ϭ
9
9
d.
1
62
result 1k 0 ϭ 12
c. 13x2 0 ϩ 3x0 ϩ 3Ϫ2 ϭ 1 ϩ 3112 ϩ
WORTHY OF NOTE
product property
a6Ϫ2 ϭ
3h7
4
ϭ4ϩ
simplify
simplify
7 0
ϭ
power property
result
Now try Exercises 27 through 66 ᮣ
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Section R.2 Exponents, Scientific Notation, and a Review of Polynomials
15
Summary of Exponential Properties
For real numbers a and b, and integers m, n, p (excluding 0 raised to a nonpositive power)
Product property:
bm # bn ϭ bmϩn
#
Power property:
1bm 2 n ϭ bm n
Product to a power:
1ambn 2 p ϭ amp # bnp
am p amp
Quotient to a power:
a n b ϭ np 1b 02
b
b
m
b
Quotient property:
ϭ bmϪn 1b 02
bn
Zero exponents:
b0 ϭ 11b 02
1 1
a Ϫn
b n
bϪn
Negative exponents:
ϭ n , Ϫn ϭ bn, a b ϭ a b 1a, b 02
a
1
b b
b
A. You’ve just seen how
we can apply properties of
exponents
B. Exponents and Scientific Notation
In many technical and scientific applications, we encounter numbers that are either
extremely large or very, very small. For example, the mass of the Moon is over 73 quintillion kilograms (73 followed by 18 zeroes), while the constant for universal gravitation contains 10 zeroes before the first nonzero digit. When computing with numbers
of this magnitude, scientific notation has a distinct advantage over the common decimal notation (base-10 place values).
WORTHY OF NOTE
Scientific Notation
Recall that multiplying by 10’s (or
multiplying by 10k, k 7 02 shifts the
decimal point to the right k places,
making the number larger. Dividing
by 10’s (or multiplying by
10Ϫk, k 7 0) shifts the decimal
point to the left k places, making
the number smaller.
A non-zero number written in scientific notation has the form
N ϫ 10k
where 1 Յ 0N 0 6 10 and k is an integer.
To convert a number from decimal notation into scientific notation, we begin by
placing the decimal point to the immediate right of the first nonzero digit (creating a
number less than 10 but greater than or equal to 1) and multiplying by 10k. Then we
determine the power of 10 (the value of k) needed to ensure that the two forms are
equivalent. When writing large or small numbers in scientific notation, we sometimes
round the value of N to two or three decimal places.
EXAMPLE 5
ᮣ
Converting from Decimal Notation to Scientific Notation
The mass of the Moon is about 73,000,000,000,000,000,000 kg. Write this number
in scientific notation.
Solution
ᮣ
Place decimal to the right of first nonzero digit (7) and multiply by 10k.
73,000,000,000,000,000,000 ϭ 7.3 ϫ 10k
To return the decimal to its original position would require 19 shifts to the right, so
k must be positive 19.
73,000,000,000,000,000,000 ϭ 7.3 ϫ 1019
The mass of the Moon is 7.3 ϫ 1019 kg.
Now try Exercises 67 and 68 ᮣ
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Converting a number from scientific notation to decimal notation is simply an
application of multiplication or division with powers of 10.
EXAMPLE 6
ᮣ
Converting from Scientific Notation to Decimal Notation
The constant of gravitation is 6.67 ϫ 10Ϫ11. Write this number in common
decimal form.
Solution
ᮣ
Since the exponent is negative 11, shift the decimal 11 places to the left, using
placeholder zeroes as needed to return the decimal to its original position:
6.67 ϫ 10Ϫ11 ϭ 0.000 000 000 066 7
Now try Exercises 69 and 70 ᮣ
Computations that involve scientific notation typically use real number properties and
the properties of exponents.
EXAMPLE 7
ᮣ
Storage Space on a Hard Drive
A typical 320-gigabyte portable hard drive can hold about 340,000,000,000 bytes
of information. A 2-hr DVD movie can take up as much as 8,000,000,000 bytes of
storage space. Find the number of movies (to the nearest whole movie) that can be
stored on this hard drive.
Solution
ᮣ
Using the ideas from Example 5, the hard drive
holds 3.4 ϫ 1011 bytes, while the DVD requires
8.0 ϫ 109 bytes. Divide to find the number of
DVDs the hard drive will hold.
3.4 ϫ 1011
3.4
1011
ϫ
ϭ
8.0
8.0 ϫ 109
109
ϭ 0.425 ϫ 102
ϭ 42.5
rewrite the
expression
divide; subtract
exponents
result
The drive will hold approximately 42 DVD movies. A calculator check is shown
in the figure.
B. You’ve just seen how
we can perform operations in
scientiﬁc notation
Now try Exercises 71 and 72 ᮣ
C. Identifying and Classifying Polynomial Expressions
A monomial is a term using only whole number exponents on variables, with no variables in the denominator. One important characteristic of a monomial is its degree. For
a monomial in one variable, the degree is the same as the exponent on the variable. The
degree of a monomial in two or more variables is the sum of exponents occurring on
variable factors. A polynomial is a monomial or any sum or difference of monomial
terms. For instance, 12x2 Ϫ 5x ϩ 6 is a polynomial, while 3nϪ2 ϩ 2n Ϫ 7 is not (the
exponent Ϫ2 is not a whole number). Identifying polynomials is an important skill because they represent a very different kind of real-world model than nonpolynomials. In
addition, there are different families of polynomials, with each family having different characteristics. We classify polynomials according to their degree and number of
terms. The degree of a polynomial in one variable is the largest exponent occurring on
the variable. The degree of a polynomial in more than one variable is the largest sum
of exponents in any one term. A polynomial with two terms is called a binomial
(bi means two) and a polynomial with three terms is called a trinomial (tri means
three). There are special names for polynomials with four or more terms, but for these,
we simply use the general name polynomial (poly means many).