Section 5.2 Properties of
Exponents and Power Functions
October 16, 2008
Warm up Lesson 01
1. Which of the related quantities,
weight of a bag of apples or
the cost of a bag of apples is
the independent variable?
4. What is the domain and
range of the following graph?
2. Sketch a graph that increases
throughout, first slowly then at
a faster rate?
5. If f(x) = x + 5 and
3. If p(x) = x2 + x + 3, what is p(3)
g(x) =
x 5
8
What is g(f(x))?
Properties of Exponents and Power functions
Frequently, you will need to rewrite a
mathematical expression in a different form to
make the expression easier to understand or
an equation easier to solve. Recall that in
exponential expression, such as 43, the
number 4 is call the base and the number 3 is
call the exponent.
If we look at 43 differently we could say
43 = 4  4  4 this is called expanded form.
Write each product in expanded form, and then rewrite it in
exponential form (with exponents).
23 
x4  x8
24
Expanded form
Expanded form
2 2  2  2  2  2  2
x x  x  x  x  x  x  x  x  x  x  x
Exponential form
27
102  105
Expanded form
10  10  10  10  10  10  10
Exponential form
107
Exponential form
x12
Generalize your results
am

an
m + n
a
= _____
Rewrite each expression in
exponential form.
#1
a5 a-2
(-5x2)  (3x5)
(-2a3b2)  (-5a5b3)
a3
-15x7
10a8b5
#4
#2
#5
#3
#6
bc5 c-4
(x5)  (x5)
(a2b2)  (-5a3b3)
bc
x10
-5a5b5
Write the numerator and denominator of each quotient in
expanded form. Reduce by eliminating common factors, and then
rewrite the factors that remain in exponential form.
5
4
43
x8
6
x
(0.94)15
(0.94) 5
44444 x x x x x x x x
xxxxxx
444
42
(0.94)10
x2
m
Now generalize your results.
a
m-n
a

an
Practice lesson 01
Rewrite in the form axn
5
5
7.
x
3
x
x2
7
10.
2x
4

x
x3
2x8
8.
6x
3x
2x4
7
x
3
(

5
x
) 4
11.
x
-5x6
9.
20 x
4x
7
5x6
2
2
ab
ba
12.
 2
ab b
a2
Write each quotient in expanded form, reduce, and rewrite in
exponential form.
3
2
2
24
4
46
44444
444444
22
2222
1
22
x
x8
5
=
2-2
xxx
xxxxxxxx
1
-5
=
x
5
x
1
= 4-1
4
Rewrite each quotient above using the property you discovered
Previously.
22-4 = 2-2 45-6 = 4-1 x3-8 = x-5
m
a
m-n
a

n
a
1
Generalize your results. n 
a
a-n
#13
#15
Rewrite each expression with
positive exponents
2
2
2
2
c ab
ab
ba
 4
2
4
#14
c ab
ab 2 b
a
2
b
6
b
4
c
3 7 3 4
f l a g
5 2 5
f la g
6
l a
2
f g
#16
c7 h 4 s2 e2
c5h 3a 3
c2ha3s2e2
Expand each of these equations and then rewrite the expression
in exponential form. Then generalize your results.
(32)4
(32) (32) (32) (32)
(53)2
(c4)2
(53) (53)
(c4) (c4)
33 3 3 3 3 3 3 55 5 5 5 5
38
c c  c  c  c  c  c  c
56
Now generalize these steps (an)m = anm
c8
Use the power rule to simplify these terms.
17.
18.
c h
b
20.
c h
3
2
b
19.
2c
6
c h
2
2 d e
4
2e
 8
d
4
3
2c
12
4
21.
c2d eh
2
8d e
3
6 3
c2 f
16 g
8
f
8
2
g
h
2 4
Property of exponents
(put on your index cards)
For a > 0, b > 0, and all values of m and n, these properties are true.
Product property of exponents
am  an = am + n
Quotient property of exp.
m
a
m-n
a

an
Zero exponents
a0 = 1
Power of a power property
(am)n = amn
Power of a product property
(ab)m = ambm
Power of a quotient property
FG IJ
HK
a
b
n
an
 n
b
Power property of equality
If a = b, then an = bn
Definition of negative exp.
Common Base property of equality
1
n
n
n
a  n or a
b
If an = am, then n = m

a
b
a
FG IJ FG IJ
HK HK
Warm up lesson 1a
𝑎𝑐 3 𝑏 2
𝑏𝑎3 𝑐 2
5
2
−2
52/3
−2
3
3𝑥
3
24 𝑥 4
2
Solve each of the following without a calculator.
Hint… use the power of a power property to convert each side of the
equation to a common base
8x
=4
27 x 
(23)X = 22
1
81
23X = 22 power of a power
c3 h
1
 4
3
3x = 2
33x
3-4
3 x
x=
2/
common base p. of =
3
1. Get a common base
2. Put in exponential form
3. Set exp = to each other
4. Solve for the variable
=
3x = -4
x = - 4/ 3
FG 49 IJ  FG 3IJ
H 9 K H 7K
x
FG 7 IJ  FG 3IJ
H 3 K H 7K
3
2
x
2
3
2
2
F FG 7 IJ I  FG 3IJ
GH H 3K JK H 7 K
2
x
3
2
F FG 7 IJ I  F FG 7 IJ I
GH H 3K JK GH H 3K JK
2
x
1
FG 7 IJ  FG 7 IJ
H 3K H 3K
2x
3
2x  
2

3
2
3
2
x = - 3/4
Practice problems 1A
1. 16x = 1/8
x = 3/4
2. 25x = 125
x = 3/2
3. 81x = 1/3
x =- 1/4
Solve for x
x4
(x4)1/4
= 3000
=
(3000)1/4
x  7.4
Hint use reverse
order of operation
6x2.5 = 90
6
6
x2.5 = 15
(x2.5)1/2.5 = (15)1/2.5
x  2.95
Practice 1a– Solve for x if needed
round to 2 decimal points
#3
x6 = 3000
x  3.80
#6
4x1/5 – 6 = 12.1
#4
#5
x-4 = 274
x1/2 = 27
x  0.25
x = 729
#7
2x-3 = 3x6
x  1897.11
x  .96
#8 answer this riddle
Mr. Norris told his girl friend , the mathematics
professor, that he would make her breakfast.
She handed him this message
( Eas) 1 (ter ) 0 Egg
I want
y
What should Mr. Norris make his girlfriend for breakfast?
1Egg
Easy
“One Egg over Easy”
Homework
• Lesson 01 all