Section 56 .ppt

Exponent Rules and Dividing
Polynomials
5.6
1. Divide exponential forms with the same base.
2. Divide numbers in scientific notation.
3. Divide monomials.
4. Divide a polynomial by a monomial.
6. Simplify expressions using rules of exponents.
5. Use long division to divide polynomials.

Objective 1
Divide exponential forms with the
same base.

Quotient Rule for Exponents
If m and n are integers and a is a real number,
where a 0, then
 .
m
m n
n
a
a
a


3
7
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
7









4
4
3
7
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2











1 1 1
1 1 1

When an equation in one variable is solved the answer is a point on a line.
Divide:
14
6
j
j
3
5
u
u
3
6
m
m
2
2
3
5
y
x
y
x
2
u 8
1
j

8
14
6 

 j
j
9
3
6 


 m
m 9
1
m
 y
x3

Objective 2
Divide numbers in scientific notation.

Divide and write the result in scientific notation.
7
3
2.34 10
3.6 10


7
3
2.34 10
3.6 10
 
7 3
0.65 10 
 
4
0.65 10
 
3
6.5 10
 
Note: This is not in scientific
notation.

Divide and write the result in scientific notation:
2
4
10
88
1
10
964
9



.
.
15
5
10
6
6
10
32
1

 
.
. 20
10
2 

.
21
10
0
2 

.
6
10
3
5 
.

Copyright © 2011 Pearson Education, Inc.
Divide.
3 6 2
4 3
18
24
x y z
x y
3 6 2
4 3
18
24 1

x y z
x y
3 4 6 3 2
18
24
 
 x y z
1 3 2
3
4

 x y z
3 2
3 1
4 1 1

y z
x
3 2
3
4

y z
x
  
  
  

1
24
18 2
3
6
4
3
z
y
y
x
x




Divide.
3 6 2
4 3
18
24
x y z
x y
3
1
24
18 2
3
6
4
3
z
y
y
x
x




4
1
x
y3
1
x
z
y
4
3 2
3


Dividing Monomials
1. Divide the coefficients or simplify them to
fractions in lowest terms.
2. Use the quotient rule for the exponents with like
bases.
3. Write the final expression so that all exponents
are positive.

Divide:
7
2
4
5
42
28
q
p
r
q
p
3
2
4
5
4
4
y
xy
xy
xy 64
43

3
3
3
2
q
r
p

Objective 4
Divide a polynomial by a monomial.

5
2
7
4
5
2
5
7
5
4 




5
2
5
7
5
4
5
2
7
4





From arithmetic…
or

Divide.
6 2 3
2
36 9 6
3
 
x y x y x
x y
6 2 3
2 2 2
36 9 6
3 3 3
  
x y x y x
x y x y x y
4 2
12 3
x y x
xy
  
Divide each term in the
polynomial by the monomial.

If a, b, and c are real numbers, variables, or
expressions with c 0, then
Dividing a Polynomial by a Monomial
Divide each term in the polynomial by the
monomial.
 .

 
a b a b
c c c

Divide:
2
2
5
3
5
15
30
xyz
xyz
yz
x


2
2
2
4
4
4
28
16
hk
k
h
hk
hk 

h
k
k 

7
4 2
3
6 3
2

 z
x

Objective 6
Simplify expressions using rules of
exponents.

Exponents Summary
Zero as an exponent: a0 = 1, where a 0.
Negative exponents:
Product rule for exponents:
Quotient rule for exponents:
Raising a power to a power:
Raising a product to a power:
Raising a quotient to a power:

1 1
, ,
n n
n n
a a
a a


 


m n m n
a a a
m
m n
n
a
a
a


  
n
m mn
a a
  
n n n
ab a b
   


n n
a b
b a
  
n
n
n
a a
b b


Use the quotient rule for exponents.
Use the power rule for exponents.
20
5
6
3
2
4
m
m

 
 5
4
3
2
2
4
m
m
14
2
m

20
6
32
64
m
m

2
1
1
m14
Evaluate coefficients.
Reduce coefficients.

4
1
2
2










z
y
x
4
4
8
4
2



z
y
x
8
4
4
4
2
x
z
y
 8
4
4
16
x
z
y


 
8
6
5
3


y
y
y
8
6
15



y
y
y
15
6
8
y
y
y
 21
8
y
y
 13
1
y


Simplify:
p
n
m
p
n
m
3
1
2
1
6
3




 
 3
7
2
4
6
3
y
y
3
2
7
3
p
n
m
13
24
1
y

24
Simplify.
a)
b)
c)
d)
2
5
3

 
 
 
a
a
16
1
a
4
1
a
16
a
4
a
5.6

25
Simplify.
a)
b)
c)
d)
2
5
3

 
 
 
a
a
16
1
a
4
1
a
16
a
4
a
5.6

26
Divide.
a) b)
c) d)
5.6
6
8
4
6
3
18
15
18
x
x
x
x 


2
5
6
6
x
x 
 2
2
6
5
6 x
x



2
6
5
6 x
x


 x
x
6
5
6 



27
Divide.
a) b)
c) d)
5.6
6
8
4
6
3
18
15
18
x
x
x
x 


2
5
6
6
x
x 
 2
2
6
5
6 x
x



2
6
5
6 x
x


 x
x
6
5
6 



Section 56 .ppt

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Section 56 .ppt