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Homework
Section 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, 45-52
2) WB pg 45 all
Section 8.4: 1) pg 393 6-37, 40-51
2) WB pg 46 all
Section 8.1
Chapter 8
Exponent Rule:
For all real numbers x and all positive integers m,
x  x  x  x  .......  xm1  xm
m
where x is the base and m is the exponent.
Examples:
25=
28=
53=
34=
Follow the sequence…2m
Section 8.1
...___...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 Remember
Any 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 Property
For all real numbers x and all integers m and n,
x x  x
m
n
mn
Expand the following to get your answer.
23 • 2 5 =
3 • 37 =
y4 • y6 =
73 • 7 n =
**See how it is easier to keep the base, and add
the exponents
Section 8.1
A) Solve for x:
2 2  2
x
3
7
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
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.
(5 y)(30 y 2 )
(4a 2b)(ac 2 )(3b 2c 2 )
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:
( x 3 )( x 5 ) 
(x ) 
4 3
First expand. What do you notice?
RULE:
•Side-by-side you ADD the exponents.
•Power-to-power you MULTIPLY the exponents
Section 8.2
Power-of-a-Power Property
For all real numbers x and all integers m and n,
(x )  x
m n
mn
Simplify and find the value of each expression when possible:
1) (23 ) 4 
3)
( p 2 )5 
2) (103 ) 2 
4)
( xm )2 
Section 8.2
Power-of-a-Product Property
For all real numbers x and y, and all integers n,
( xy) n  x n y n
Simplify:
1)
2
3
3)
2) (ab 2c n )5 
4) (3ab 2 c 6 ) 4 
( x y) 
(5xy3 ) 2 
Section 8.2
When is the negative sign apart of the base?
( x) 3 compared to  x 3
(2 x3 y) 4 compared to  (2 x 3 y) 4
Rule:
•Even power with a negative sign….answer is positive
•Odd power with a negative sign….answer is negative.
1)
(5 x)
2)
 5x
4
4
3)
(2 x)
4)
 ( 2 x) 3
3
Section 8.3
Expand the following. What could a rule be?
26

4
2
x3

7
x
x6 y 4 z

3
3
x yz
Section 8.3
Quotient-of-Powers Property
For all nonzero real numbers x and all integers m and n,
xm
mn

x
xn
Simplify:
9
1) 10

2
10
9 4
x
y z
2)

2 4 5
x y z
10 x 2 y 5 z
3)

3
 2x y z
4)
4c 4 b

2
20c a
Section 8.3
Simplify:
1)
xa

x
2)
x a b

c
x
3)
x m 1

x
Section 8.3
Power-of-a-Fraction Property
For all real numbers a and b where b ≠ 0 , and all integers n
a n an
( )  n
b
b
Simplify:
3 2
1) ( ) 
4
3 2
3) ( ) 
4
10 4
2) ( ) 
5
x3 z
4) ( y ) 
2
Follow the sequence…2m
Section 8.4
…___...___...___...___...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
Definition of Negative Exponent
For all nonzero real numbers x and all integers n,
x
n
1
 n
x
Simplify:
3
2
1) 2  2 
3)
103
2)

1
10
4)
4

2
3
x 2

5
x
Section 8.4
Simplify:
1)  3y
2

4 5
4
c
d
2)

4 8
12c d
3) m
n
2
3

6
2
b

b
4)

4
b
5) c 8 d 3 
5
4
(
2
a
)(
10
a
)
6)

3
4a