HomeworkHomeworkSection 8.1: 1) pg 374 6-17, 30-37, 44-51, 60

2) WB pg 43 all

Section 8.2: 1) pg 381 1, 4, 6-13, 18-25, 26-49, 54-64

2) WB pg 44 all

Section 8.3: 1) pg 387 1, 2, 5-36, 45-522) WB pg 45 all

Section 8.4: 1) pg 393 6-37, 40-512) WB pg 46 all

Chapter 8Chapter 8

Exponent Rule:For all real numbers x and all positive integers

m,

where x is the base and m is the exponent.

Section 8.1

mmm xxxxxx 1.......

Examples:

25= 28=

53= 34=

Follow the sequence…2m

...___...2 …4 …8…16…

…20...21…22…23…24…

Exponents mean you start with the base, and keep multiplying by that base. Working from right to left…you would divide by that base. So there are rules for x1, x0 (when x ≠ 0). What would those rules be?

Section 8.1

Exponent Facts to Exponent Facts to RememberRememberAny base to the power of 1 equals the

base.Ex: 81 =Ex: x1 =

Any base to the power 0 equals 1.Ex: 80 =Ex: x0 =

When a base is 1 the answer is always 1.Ex: 16 =

Section 8.1

Product-of-Powers PropertyFor all real numbers x and all integers m

and n, nmnm xxx Expand the following to get your answer.23 • 25 =

3 • 37 =

y4 • y6 =

73 • 7n =

**See how it is easier to keep the base, and add the exponents

Section 8.1

A) Solve for x:

x = ______

B) Suppose a colony of bacteria doubles in size every hour. If the colony contains 1000 bacteria at noon, how many bacteria will the colony contain at 3 p.m. and at 5 p.m. of the same day.

Either make a table…or use exponents

73 222 x

Section 8.1

The Product-of-Powers Property can be used to find the product of more complex expressions such as 5a2b and –2ab3. Expressions like 5a2b and –2ab3 are called monomials.

Definition of Monomial

A monomial is an algebraic expression that is either a constant, a variable, or a product of a constant and one or more variables. The constant is called the coefficient.

Section 8.1

)30)(5( 2yy )3)()(4( 2222 cbacba

Section 8.1

The volume, V, of a right rectangular prism can be found by using the formula V = lwh. Suppose that a prism has a length of 2xy, a width of 3xy, and a height of 6xyz. Find the volume.

HW: WB pg 43. Do 1, 13, 21, 31 together if time:

Section 8.2

Multiplying monomials side-by-side is different than raising power-to-power.

Side-by-side:

Power-to-Power:

First expand. What do you notice?

RULE: •Side-by-side you ADD the exponents.•Power-to-power you MULTIPLY the exponents

))(( 53 xx

34 )(x

Section 8.2

Power-of-a-Power Property

For all real numbers x and all integers m and n,

mnnm xx )(

Simplify and find the value of each expression when possible:

1) 3)

2) 4)

43)2(

23)10(

52 )( p

2)( mx

Section 8.2

Power-of-a-Product Property

For all real numbers x and y, and all integers n,

nnn yxxy )(

Simplify:1) 3)

2) 4)

32 )( yx

52 )( ncab

23)5( xy

462 )3( cab

Section 8.2

When is the negative sign apart of the base?

compared to

compared to

3)( x 3x

43 )2( yx 43 )2( yx

1) 3)

2) 4)

4)5( x

45x

3)2( x

3)2( x

Rule:•Even power with a negative sign….answer is positive•Odd power with a negative sign….answer is negative.

Section 8.3

4

6

2

2

Expand the following. What could a rule be?

7

3

x

x

33

46

yzx

zyx

Section 8.3

Quotient-of-Powers Property

For all nonzero real numbers x and all integers m and n,

nmn

m

xx

x

Simplify:

1) 3)

2) 4)

2

9

10

10

542

49

zyx

zyx

ac

bc2

4

20

4

zyx

zyx3

52

2

10

Section 8.3

Simplify:

1)

2)

3)

x

xa

c

ba

x

x

x

xm 1

Section 8.3

Power-of-a-Fraction Property

For all real numbers a and b where b ≠ 0 , and all integers n

n

nn

b

a

b

a)(

Simplify:

1) 3)

2) 4)

2)4

3(

4)5

10(

2)4

3(

zy

x)

2(

3

Follow the sequence…2m

…___...___...___...___...2 …4 …8…16…

…___...___...___...___...21…22…23…24…

Exponents mean you start with the base, and keep multiplying by that base. Working from right to left…you would divide by that base. Notice what happens when you get to negative exponents.

Section 8.4

Section 8.4

Definition of Negative Exponent

For all nonzero real numbers x and all integers n,

nn

xx

1

Simplify:

1) 3)

2) 4)

23 22

1

3

10

10

23

4

5

2

x

x

Section 8.4

Simplify:

1) 4)

2) 5)

3) 6)

23y

84

54

12

4

dc

dc

3

2

n

m

4

26

b

bb

38dc

3

45

4

)10)(2(

a

aa