objective 1: to multiply monomials. objective 2: to divide monomials and simplify expressions with...

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