objective 1: to multiply monomials. objective 2: to divide monomials and simplify expressions with...
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Objective 1: To multiply monomials.Objective 2: To divide monomials and
simplify expressions with negative exponents.
To multiply monomials
Exponent ReviewExponent Review
BaseBase
ExponenExponentt52
2 2 2 2 2
It means to use 2 as a factor 5 times.
52
8 2 2
52 2 2 2 2 2 4 2 2 2
16 232
52 32
Exponent ReviewExponent Review
BaseBase
ExponenExponentt
3( 4)
( 4)( 4)( 4)
64
3( 4)16( 4)
643( 4)
Multiplying with exponentsMultiplying with exponents When you multiply, add the exponents.
3 5 8x x x 14 5y y y
Example 1Example 15 3 21( 4 )(3 )x y x y
What’s the exponent for this x?What’s the exponent for this x?
Example 1Example 15 3 21( 4 )(3 )x y x y
Multiply the coefficients first.Multiply the coefficients first.
Example 1Example 15 3 21( 4 )(3 )x y x y
6 512x y
Example 1Example 15 3 21( 4 )(3 )x y x y
6 512x y
Example 2Example 22 3 3 5(5 )(2 )x y x y
5 810x y
Example 2Example 22 3 3 5(5 )(2 )x y x y
5 810x y
Raising a power to a powerRaising a power to a powerWhen raising a power to another
power, multiply the exponents.
2 3 6( )a a4 5 20( )x x
Raising a power to a powerRaising a power to a powerWhen raising a power to another
power, multiply the exponents.
3 4 2 6 8( )x y x y
Take everything in parentheses and raise it to the 2Take everything in parentheses and raise it to the 2ndnd power. power.
Raising a power to a powerRaising a power to a powerWhen raising a power to another
power, multiply the exponents.
2 5 5( 2) 5 10( )n n nx x x Use Distributive Property.Use Distributive Property.
Example 3Example 33 4 3(2 )a b
3 9 122 a b
Simplify the coefficient.Simplify the coefficient.
Example 3Example 33 4 3(2 )a b
3 9 122 a b9 128a b
2 2 2 8
2 4 31 1(2 )( 3 )x y x yExample 4Example 4
What’s the exponent for this What’s the exponent for this x?x?
What’s the exponent for this What’s the exponent for this x?x?
2 4 31 1(2 )( 3 )x y x yExample 4Example 4
Simplify this expression first Simplify this expression first because it has an outside exponent.because it has an outside exponent.
Take everything in parentheses and Take everything in parentheses and raise it to the 3raise it to the 3rdrd power. power.
2 4 31 1(2 )( 3 )x y x yExample 4Example 4
1 2 3 12(2 )( 27 )x y x yNow, there are no outside exponents.Now, there are no outside exponents.
We can multiply the coefficientsWe can multiply the coefficients
Then multiply the x’sThen multiply the x’s
Then multiply the y’sThen multiply the y’s
( 3)( 3)( 3) 27
Example 4Example 4
1 2 3 12(2 )( 27 )x y x y4 1454x y
2 4 31 1(2 )( 3 )x y x y
Example 4Example 4
1 2 3 12(2 )( 27 )x y x y4 1454x y
2 4 31 1(2 )( 3 )x y x y
Example 5 (skip)Example 5 (skip)2 2 3[(2 ) ]xy
Start outside the bracketsStart outside the brackets
Multiply the two outside Multiply the two outside exponents.exponents.
Example 5Example 5
2 6(2 )xy
2 2 3[(2 ) ]xy
Simplify the outside exponent.Simplify the outside exponent.
Raise everything inside the Raise everything inside the parentheses to the 6parentheses to the 6thth power. power.
Example 5Example 5
2 6(2 )xy
2 2 3[(2 ) ]xy
6 6 122 x y6 1264x y
To divide monomials and simplify negative exponents.
Dividing with exponentsDividing with exponents When you divide, subtract the exponents.
52
3
22 4
2
102
8
xx
x
Example 6Example 6
1
5 9
4
a b
a b
What’s the exponent for this b?What’s the exponent for this b?
Example 6Example 6
1
5 9
4
a b
a b1 8a b
8ab
Example 7Example 723
2
x
x
Raise everything in the parentheses Raise everything in the parentheses to the 2to the 2ndnd power. power.
Example 7Example 723
2
x
x
6
4
x
xSubtract the exponents.Subtract the exponents.
Example 7Example 723
2
x
x
6
4
x
x2x
Example 8Example 832
4
y
Raise everything in the parentheses Raise everything in the parentheses to the 3to the 3rdrd power. power.
Example 8Example 832
4
y
6
34
y
Simplify the denominator.Simplify the denominator.
4 4 4 64
Example 8Example 832
4
y
6
34
y
6
64
y
4
41
x x x x x
x x x x x
Look At ThisLook At This4
04
xx
x
1
1
1
1
1
1
1
1
4
41
x x x x x
x x x x x
Look At ThisLook At This4
04
xx
x
Look At ThisLook At This
0 1x This is another rule.This is another rule.
Zero as exponentZero as exponent Anything raised to zero power equals 1.
0 1y
4 3 0(7 ) (1) 1x y
09 1
Review: Whole NumbersReview: Whole Numbers
4
Any whole number can be placed on top of 1.Any whole number can be placed on top of 1.
4
1
Review: Whole NumbersReview: Whole Numbers
7Any whole number can be placed on top of 1.Any whole number can be placed on top of 1.
7
1
Review: Whole NumbersReview: Whole Numbers
15
Any whole number can be placed on top of 1.Any whole number can be placed on top of 1.
15
1
Review: Whole NumbersReview: Whole Numbers
x
Any whole number can be placed on top of 1.Any whole number can be placed on top of 1.
1
x
Review: Whole NumbersReview: Whole Numbers
yAny whole number can be placed on top of 1.Any whole number can be placed on top of 1.
1
y
Review: Whole NumbersReview: Whole Numbers
5x
Any whole number can be placed on top of 1.Any whole number can be placed on top of 1.
5
1
x
FractionsFractionsThere are 2 parts of a fraction.There are 2 parts of a fraction.
top
bottom
Negative ExponentsNegative ExponentsWhen you see negative exponents, thinkWhen you see negative exponents, think
MOVE & CHANGEMOVE & CHANGE
MoveMove the base from top to bottom or bottom the base from top to bottom or bottom to top.to top.
ChangeChange the exponent to a positive number. the exponent to a positive number.
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
4 9
9 4
x y
y x
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
4 9
9 4
x y
y x
3
2 3 2
1x
y x y
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGEy does not have negative exponent.y does not have negative exponent.
It stays where it is.It stays where it is.
Nothing is left on top.Nothing is left on top.
We know there is an invisible 1 there.We know there is an invisible 1 there.
3
2 3 2
1x
y x y
Negative ExponentsNegative Exponents
44
4
2 1 12
1 2 16
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
44
4
2 1 12
1 2 16
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
33
3
1
1
xx
x
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
33
3
1
1
xx
x
Negative ExponentsNegative Exponents
MOVE & CHANGEMOVE & CHANGE
Example 9Example 922
3
a
b
Raise everything in the parentheses Raise everything in the parentheses to the negative 2to the negative 2ndnd power. power.
Example 9Example 922
3
a
b
4
6
a
b
Move the negative exponents and Move the negative exponents and change to positive exponents.change to positive exponents.
Example 9Example 922
3
a
b
4
6
a
b
6
4
b
a
Example 10Example 1057d
57
1
d
Example 10Example 1057d
5
7
1d
5
7
d
Example 11Example 11
4
2
5a
Move any negative exponents and Move any negative exponents and change to positive.change to positive.
Example 11Example 11
4
2
5a
42
5
a
Example 11Example 11
4
2
5a
42
5
a
Example 12 (skip)Example 12 (skip)4
9
x
xSubtract the exponents.Subtract the exponents.
4 9 5
Example 12Example 124
9
x
x5x
Put under 1 and change exponent to Put under 1 and change exponent to positive.positive.
Example 12Example 124
9
x
x5x
5
1
x
The variable stays where the bigger exponent was.
IMPORTANT!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 FormAll Rules in Symbolic Form
0
1
m n m n
mm n
n
x x x
xx
x
x
( )
m n mn
pm mp
n np
pm n mp np
x x
x x
y y
x y x y
1nn
xx
Example 13Example 134 3 7(3 )( 2 )ab a b
2 36a b1 ( 3) 2
4 7 3
Example 13Example 134 3 7(3 )( 2 )ab a b
2 36a b
Move negative exponents and Move negative exponents and change to positive.change to positive.
Example 13Example 134 3 7(3 )( 2 )ab a b
2 36a b3
2
6b
a
Example 13Example 134 3 7(3 )( 2 )ab a b
2 36a b3
2
6b
a
Example 14Example 142 5
5 2
4
6
a b
a b
7 32
3
a b
2 5 7 5 2 3
Example 14Example 142 5
5 2
4
6
a b
a b
7 32
3
a b
Move negative exponents and Move negative exponents and change to positive.change to positive.
Example 14Example 142 5
5 2
4
6
a b
a b
7 32
3
a b
3
7
2
3
b
a
Example 14Example 142 5
5 2
4
6
a b
a b
7 32
3
a b
3
7
2
3
b
a