copyright © 2010 pearson education, inc. all rights reserved. 5.5 – slide 1


Exponents and Polynomials

Chapter 5


5.5

Integer Exponents and the Quotient Rule


Objectives

1. Use 0 as an exponent.2. Use negative numbers as exponents.3. Use the quotient rule for exponents.4. Use combinations of rules.

5.5 Integer Exponents and the Quotient Rule


Zero ExponentFor any nonzero real number a,

a

0 = 1.

Example: 170 = 1

5.5 Integer Exponents and the Quotient RuleUsing 0 as an Exponent


(a) 380

Example 1 Evaluate.

5.5 Integer Exponents and the Quotient RuleUsing 0 as an Exponent

(b) (–9)0

(c) –90 = –1(9)0 = –1(1)= –1

(d) x0 = 1

= 1

= 1

(e) 5x0 = 5·1= 5

(f) (5x)0 = 1


Negative ExponentsFor any nonzero real number a and any integer n,

Example:

5.5 Integer Exponents and the Quotient RuleUsing Negative Numbers as Exponents

a n

1

an.

3 2

1

32

1

9.


Example 2 Simplify by writing with positive exponents. Assume that allvariables represent nonzero real numbers.


(a) 9–33

1

9

1

729

31

(b) 4

34

1

64Notice that we can change the base to its reciprocal if we also change the sign of the exponent.

52

(c) 3

53

2

243

32


Example 2 (concluded)Simplify by writing with positive exponents. Assume that allvariables represent nonzero real numbers.


1 1(d) 6 3 1 1

6 3

1 2

6 6

1

6

4

3(e)

x

4

31x

4

4

x

x

43x

= 1


CAUTIONA negative exponent does not indicate a negative number. Negative exponents lead to reciprocals.


33

1 12

2 8

Expression Example

a–n Not negative

–a–n 33

1 12

2 8 Negative



Changing from Negative to Positive ExponentsFor any nonzero numbers a and b and any integers m and n,

a m

b n

bn

am and

a

b

m

b

a

m

.

Examples:3 35 4

4 5

3 2 4 5 and .

2 3 5 4


CAUTIONBe careful. We cannot use the rule tochange negative exponents to positive exponents if theexponents occur in a sum or difference of terms. For example,


2 1

3

5 3

7 2

would be written with positive exponents as2

3

1 15 3 .

17

2

m n

n m

a b

b a


5.5 Integer Exponents and the Quotient RuleUsing the Quotient Rule for Exponents

Quotient Rule for ExponentsFor any nonzero number a and any integers m and n,

.m

m nn

aa

a

Example:8

8 6 26

55 =5 =25.

5

(Keep the same base and subtract the exponents.)


CAUTIONA common error is to write This is incorrect. By the quotient rule, the quotient must have the same base, 5, so


We can confirm this by using the definition of exponents to write out the factors:

Using the Quotient Rule for Exponents

58

5618 6 12.

58

5658 6 =52.

58

56

55555555555555

.


Example 3 Simplify. Assume that all variables represent nonzero real numbers.


4

6

3(a)

34 63 2

1

323

4

9(b)

y

y

4 ( 9)y 5y4 9y


4 ( 5) 7 62 ( )z a

Example 3 (continued)Simplify. Assume that all variables represent nonzero real numbers.


4 7

5 6

2 ( )(c)

2 ( )

z a

z a

4 7

5 6

2 ( )

2 ( )

z a

z a

92 ( )z a


Example 3 (concluded) Simplify. Assume that all variables represent nonzero real numbers.


3 8

2 4 6

5(d)

3

x y

x y

3 8

2 4 6

5

3

x y

x y

2 3 4 8 65 3 x y

7 25 9x y 2

7

45y

x



Definitions and Rules for ExponentsFor any integers m and n:

Product rule am · an = am+n

Zero exponent a0 = 1 (a ≠ 0)

Negative exponent

Quotient rule


a n

1

an

am

anam n (a 0)



Definitions and Rules for Exponents (concluded)For any integers m and n:Power rules (a) (am)n = amn

(b) (ab)m = ambm

(c)

Negative-to-PositiveRules


a

b

m

am

bm(b 0)

a

b

m

b

a

m

a m

b n

bn

am(a,b 0)


Example 4Simplify each expression. Assume all variables represent nonzero real numbers.

5.5 Integer Exponents and the Quotient RuleUsing Combinations of Rules

3 2

6

(2 )(a)

2

6 62

1

6

6

2

2

02


Example 4 (continued)Simplify each expression. Assume all variables represent nonzero real numbers.


4 2

1

(3 ) (3 )(b)

(3 )

y y

y

4 2

1

(3 )

(3 )

y

y

6 ( 1)(3 )y 7(3 )y

7 73 y

72187y


Example 4 (concluded)Simplify each expression. Assume all variables represent nonzero real numbers.


23

1 4

5(c)

2

a

b

21 4

3

2

5

b

a

23 4

12 5

a b

6 8

2(10)

a b

6 8

100

a b



NoteSince the steps can be done in several different orders, there are many equally correct ways to simplify expressions like those in Example 4.

Using Combinations of Rules

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