Section 12.2 Integer Exponents

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Exponents and
Polynomials
12

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Use 0 as an exponent.
2. Use negative numbers as exponents.
3. Use the quotient rule for exponents.
4. Use combinations of the rules for
exponents.
Objectives
12.2 Integer Exponents and the Quotient
Rule

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Zero Exponent
For any nonzero real number a,
a0 = 1.
Example: 170 = 1
Use 0 as an Exponent

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
(a) 380
Example Evaluate.
Use 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

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Negative Exponents
For any nonzero real number a and any integer n,
Example:
Use Negative Numbers as Exponents
an

1
an
.
32

1
32

1
9
.

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
Use Negative Numbers as Exponents
(a) 9–3
3
1
9

1
729

3
1
(b)
4

 
 
 
3
4
1
 
  
 
64
Notice that we can change the
base to its reciprocal if we also
change the sign of the exponent.
5
2
(c)
3

 
 
 
5
3
2
 
  
 
243
32


7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
Use Negative Numbers as Exponents
1 1
(d) 6 3 

1 1
6 3
 
1 2
6 6
 
1
6
 
4
3
(e)
x
4
3
1
x

4
3x

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
CAUTION
A negative exponent does not indicate a negative
number. Negative exponents lead to reciprocals.
Use Negative Numbers as Exponents
3
3
1 1
2
2 8

 
Expression Example
a–n Not negative
–a–n 3
3
1 1
2
2 8

    Negative

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Use Negative Numbers as Exponents
Changing from Negative to Positive Exponents
For 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


   
    
   

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
CAUTION
Be careful. We cannot use the rule to
change negative exponents to positive exponents if the
exponents occur in a sum or difference of terms. For
example,
Use Negative Numbers as Exponents
2 1
3
5 3
7 2
 



would be written with positive exponents as
2
3
1 1
5 3 .
1
7
2


m n
n m
a b
b a




11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Use the Quotient Rule for Exponents
Quotient Rule for Exponents
For any nonzero number a and any integers m and n,

 .
m
m n
n
a
a
a
Example:
8
8 6 2
6
5
5 =5 =25.
5


(Keep the same base and subtract the exponents.)

12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
CAUTION
A 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:
Use the Quotient Rule for Exponents
58
56
 186
 12
.
58
56
 586
=52
.
58
56

55555555
555555
.

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Simplify. Assume that all variables represent nonzero real
numbers.
Use the Quotient Rule for Exponents
4
6
3
(a)
3
4 6
3 
 2
1
3
2
3

4
9
(b)
y
y


4 ( 9)
y  
 5
y4 9
y 


14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
4 ( 5) 7 6
2 ( )z a  
  
Example (cont)
Simplify. Assume that all variables represent nonzero real
numbers.
Use the Quotient Rule for Exponents
4 7
5 6
2 ( )
(c)
2 ( )
z a
z a


4 7
5 6
2 ( )
2 ( )
z a
z a

 

9
2 ( )z a  

15. Slide - 15Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Simplify. Assume that all variables represent nonzero real
numbers.
Use the Quotient Rule for Exponents
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 6
5 3 x y  
   
7 2
5 9x y
 
2
7
45y
x


16. Slide - 16Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Definitions and Rules for Exponents
For any integers m and n:
Product rule am · an = am+n
Zero exponent a0 = 1 (a ≠ 0)
Negative exponent
Quotient rule
Use the Quotient Rule for Exponents
an

1
an
am
an
 amn
(a  0)

17. Slide - 17Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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-Positive
Rules
Use the Quotient Rule for Exponents
a
b




m

am
bm
(b  0)
a
b




m

b
a




m
am
bn

bn
am
(a,b  0)

18. Slide - 18Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Simplify each expression. Assume all variables represent
nonzero real numbers.
Use Combinations of Rules
3 2
6
(2 )
(a)
2
6 6
2 

1
6
6
2
2

0
2

19. Slide - 19Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Simplify each expression. Assume all variables represent
nonzero real numbers.
Use Combinations of Rules
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 7
3 y
7
2187y

20. Slide - 20Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Simplify each expression. Assume all variables represent
nonzero real numbers.
Use Combinations of Rules
23
1 4
5
(c)
2
a
b


 
 
 
21 4
3
2
5
b
a


 
  
  23 4
1
2 5
a b 
  
 
6 8
2
(10)
a b

6 8
100
a b
