Transcript mult Divide monomials

Objective 1: To multiply monomials.
Objective 2: To divide monomials and
simplify expressions with
negative exponents.
To multiply monomials
Exponent Review
5
2
Base
Exponent
What does it mean?
5
2
 22222
It means to use 2 as a factor 5 times.
What does it mean?
5
2
 22222
4222
8 2 2
16  2
32
2  32
5
Exponent Review
Exponent
(4)
Base
3
What does it mean?
(4)  (4)(4)(4)
3
16(4)
64
What does it mean?
(4)  64
3
Multiplying with exponents
 When you multiply, add the exponents.
5
x x  x
8
y y  y
5
3
4
1
Example 1
(4 x y )(3x y )
5
3
1
2
What’s the exponent for this x?
Example 1
(4 x y )(3x y )
5
3
1
2
Multiply the coefficients first.
Example 1
(4 x y )(3x y )
5
3
1
12x y
6
5
2
Example 1
(4 x y )(3x y )
5
3
1
12x y
6
5
2
Example 2
2
3
3
5
(5x y )(2 x y )
5
10x y
8
Example 2
2
3
3
5
(5x y )(2 x y )
5
10x y
8
Raising a power to a power
 When raising a power to another power,
multiply the exponents.
(a )  a
6
(x )  x
20
2 3
4 5
Raising a power to a power
 When raising a power to another power,
multiply the exponents.
(x y )  x y
3
4 2
6
8
Take everything in parentheses and raise it to the 2nd power.
Raising a power to a power
 When raising a power to another power,
multiply the exponents.
(x
n2 5
) x
5( n  2)
x
Use Distributive Property.
5 n 10
Example 3
3 4 3
(2a b )
3
9 12
2 ab
Simplify the coefficient.
Example 3
3 4 3
(2a b )
2 2 2  8
3
9 12
2 ab
9 12
8a b
Example 4
(2 x y )( 3x y )
1
2
1
4 3
What’s the exponent forWhat’s
this x? the exponent for this x?
Example 4
(2 x y )( 3x y )
1
2
1
4 3
Simplify this expression first
because it has an outside exponent.
Take everything in parentheses and
raise it to the 3rd power.
Example 4
(2 x y )( 3x y )
1 2
3 12
(2 x y )(27 x y )
1
2
1
4 3
(3)(3)(3)  27
Now, there are no outside exponents.
We can multiply the coefficients
Then multiply the x’s
Then multiply the y’s
Example 4
(2 x y )( 3x y )
1 2
3 12
(2 x y )(27 x y )
1
2
1
54x y
4
14
4 3
Example 4
(2 x y )( 3x y )
1 2
3 12
(2 x y )(27 x y )
1
2
1
54x y
4
14
4 3
Example 5 (skip)
2 2 3
[(2 xy ) ]
Start outside the brackets
Multiply the two outside
exponents.
Example 5
2 2 3
[(2 xy ) ]
2 6
(2 xy )
Simplify the outside exponent.
Raise everything inside the
parentheses to the 6th power.
Example 5
2 2 3
[(2 xy ) ]
2 6
(2 xy )
6
6 12
2 x y
6 12
64x y
To divide monomials and simplify negative
exponents.
Dividing with exponents
 When you divide, subtract the exponents.
5
2
2

2

4
3
2
10
x
2

x
8
x
Example 6
5 9
ab
4 1
ab
What’s the exponent for this b?
Example 6
5 9
ab
4 1
ab
1 8
ab
ab
8
Example 7
x 
 2
x
 
3
2
Raise everything in the parentheses
to the 2nd power.
Example 7
x 
 2
x
 
6
x
4
x
3
2
Subtract the exponents.
Example 7
x 
 2
x
 
6
x
4
x
3
x
2
2
Example 8
y 
 
4
 
2
3
Raise everything in the parentheses
to the 3rd power.
Example 8
y 
 
4
 
6
y
3
4
2
3
Simplify the denominator.
4  4  4  64
Example 8
y 
 
4
 
6
y
3
4
6
y
64
2
3
Look At This
4
x
0

x
4
x
1
1
1
1
1
1
1
1
x
xxxx


1
4
x
xxxx
4
Look At This
4
x
0

x
4
x
x
xxxx


1
4
x
xxxx
4
Look At This
x 1
0
This is another rule.
Zero as exponent
 Anything raised to zero power equals 1.
y 1
0
9 1
0
(7 x y )  (1)  1
4
3 0
Review: Whole Numbers
Any whole number can be placed on top of 1.
4
4
1
Review: Whole Numbers
Any whole number can be placed on top of 1.
7
7 
1
Review: Whole Numbers
Any whole number can be placed on top of 1.
15
15 
1
Review: Whole Numbers
Any whole number can be placed on top of 1.
x
x
1
Review: Whole Numbers
Any whole number can be placed on top of 1.
y
y
1
Review: Whole Numbers
Any whole number can be placed on top of 1.
5x
5x 
1
Fractions
There are 2 parts of a fraction.
top
bottom
Negative Exponents
When you see negative exponents, think
MOVE & CHANGE
Move the base from top to bottom or bottom
to top.
Change the exponent to a positive number.
Negative Exponents
4
9
x
y

9
4
y
x
MOVE & CHANGE
Negative Exponents
4
9
x
y

9
4
y
x
MOVE & CHANGE
Negative Exponents
3
x
1

2
3 2
y
x y
y does not
Nothing
have is
negative
left on top.
exponent.
MOVE & CHANGE
We knowItthere
staysiswhere
an invisible
it is. 1 there.
Negative Exponents
3
x
1

2
3 2
y
x y
Negative Exponents
4
2
1
1
2 
 4 
1
2
16
4
MOVE & CHANGE
Negative Exponents
4
2
1
1
2 
 4 
1
2
16
4
MOVE & CHANGE
Negative Exponents
3
x
1
x 
 3
1
x
3
MOVE & CHANGE
Negative Exponents
3
x
1
x 
 3
1
x
3
MOVE & CHANGE
Example 9
a 
 3
b
 
2
2
Raise everything in the parentheses
to the negative 2nd power.
Example 9
a 
 3
b
 
4
a
6
b
2
2
Move the negative exponents and
change to positive exponents.
Example 9
a 
 3
b
 
4
a
6
b
6
b
4
a
2
2
Example 10
7d
5
Example 10
7d
5
5
7d

1
7
 5
1d
7
 5
d
Example 11
2
4
5a
Move any negative exponents and
change to positive.
Example 11
2
4
5a
4
2a
5
Example 11
2
4
5a
4
2a
5
Example 12 (skip)
4
x
9
x
Subtract the exponents.
4  9  5
Example 12
4
x
9
x
x
5
Put under 1 and change exponent to
positive.
Example 12
4
x
9
x
x
5
1
5
x
The variable
stays where
the bigger
exponent
was.
IMPORTANT!
 Your final answer can NOT have any negative
exponents.
 Remember to move all negative exponents
and change them to positives.
All Rules in Symbolic Form
x x  x
m
n
mn
m
x
mn

x
n
x
0
x 1
x
n
1
 n
x
m n
(x )  x
mn
p
mp
x  x
 n   np
y
y 
m
x y 
m
n
p
x y
mp
np
Example 13
4
3 7
(3ab )(2a b )
2 3
6a b
1  (3)  2
4  7  3
Example 13
4
3 7
(3ab )(2a b )
2 3
6a b
Move negative exponents and
change to positive.
Example 13
4
3 7
(3ab )(2a b )
2 3
6a b
6b
2
a
3
Example 13
4
3 7
(3ab )(2a b )
2 3
6a b
6b
2
a
3
Example 14
2 5
4a b
5 2
6a b
7 3
2a b
3
2  5  7
52  3
Example 14
2 5
4a b
5 2
6a b
7 3
2a b
3
Move negative exponents and
change to positive.
Example 14
2 5
4a b
5 2
6a b
7 3
2a b
3
3
2b
7
3a
Example 14
2 5
4a b
5 2
6a b
7 3
2a b
3
3
2b
7
3a