Mat 092 section 12.2 integer exponents

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

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

Zero Exponent
For any nonzero real number a,
a0 = 1.
Example: 170 = 1
Use 0 as an Exponent

(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

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
.

Example
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
(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


Example (cont)
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
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

CAUTION
A negative exponent does not indicate a negative
number. Negative exponents lead to reciprocals.
3
3
1 1
2
2 8

 
Expression Example
a–n Not negative
–a–n 3
3
1 1
2
2 8

    Negative

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


   
    
   

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,
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




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.)

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:
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
.

Example
Simplify. Assume that all variables represent nonzero real
numbers.
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 


4 ( 5) 7 6
2 ( )z a  
  
Example (cont)
numbers.
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  

Example (cont)
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 6
5 3 x y  
   
7 2
5 9x y
 
2
7
45y
x


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
an

1
an
am
an
 am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-Positive
Rules
a
b




m

am
bm
(b  0)
a
b




m

b
a




m
am
bn

bn
am
(a,b  0)

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

Example (cont)
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

Example (cont)
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


Mat 092 section 12.2 integer exponents

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