Elementary Algebra
Section 5.2
Page 1 of 3
Section 5.2: Integer Exponents and the Quotient Rule
Big Idea: There are shortcut formulas to doing calculations with exponents. These rules help us simplify
expressions to make equation solving easier.
Rules for Exponents:
For positive integers m and n and any nonzero real numbers a and b:
New Rules
Rule Name
Zero Exponent Definition
Negative Exponent Definition
Negative to Positive Rules
Quotient Rule
Rule Name
Product Rule
Rule
a0  1
1
an
bn
 m
a
an 
am
bn
m
m
a
b
   
b
a
m
a
 a mn
n
a
Old Rules
Rule
a m  a n  a mn
5
Example
6 2  6 4  6 2  4  66
 a mn
3 
 ab 
 a mb m
2 p
m
m
am
a

 
bm
b
5
 2
 3
   
 3
 2
8
2
 283  25
3
2
a 
m n
Power Rules
Example
170 = 1; (-4)0 = 1
1 1
32  2 
3
9
5
2
3
4
 5
2
4
3
2 4
5
 32(4)  38
 25 p 5
2
2
5 5

 
2
 3 3
1. Zero and Negative Exponent Activity:
1
a. Calculate 8  
8
b. Now re-write the problem with every number written using zero and positive exponents:
c. Now re-write the problem with every number written using negative, zero, and positive
exponents:
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra
Section 5.2
Page 2 of 3
d. Why are negative and zero exponents defined as they are?
2. Another Zero and Negative Exponent Activity:
a. Consider the following list:
34 = 81
33 = 27
32 = 9
b. What is the pattern going down the list (i.e., what do you have to do to a number on the right
hand side to get to the next number down)?
c. Continue the list above using this pattern
d. Why are negative and zero exponents defined as they are?
3. Quotient Rule Activity
1024

a. Calculate
128
b. Now re-write the problem with every number written using exponents:
c. How are the exponents related?
d. State the Quotient Rule in your own words.
e. Calculate
390, 625
using the Quotient Rule:
125
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra
Section 5.2
Page 3 of 3
4. Negative to Positive Rule Activities
3
 4
a. Simplify   
7
b. Simplify
32

3 5
c. Simplify
5 7

4 3
d.
4
Simplify
1
a 1b 2 
 3a
 5a
b 
2
3 5 2

3 4 2
b

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.