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A. The Properties of Exponents

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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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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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).



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