Section 4.2: Integer Exponents And The Quotient Rule
In this section we look at terms that have integer exponents. Remember, an integer is a positive and negative
whole number, so now will be examining terms that have negative exponents.
§1 The Quotient Rule
Before we go any further we should actually mention that zero is also an integer. That means an exponent can
be zero as well! What does that mean? How can we understand what a negative or zero exponent is? Well, we
am
 a m  n . It states that if we are dividing bases
an
that are the same, then we can combine the terms into a single base and subtract the exponents. Hence,
sort of need to quotient rule first. The quotient rule says that
something like
x6
58
62
4

x

x
 53 .
,
and
x2
55
§2 Zero As An Exponent
What about something like
45
? Well, the bases are the same, so the quotient rule tells us that we can combine
45
the bases and subtract the exponents. But what happens? We see that we end up with
also note that anything divided by itself is equal to 1! Hence,
45
 455  40 ! But we
5
4
45
= 40  1! This is something you need to
5
4
remember. Anything to the zero power is simply equal to 1.
PRACTICE
1) Simplify 70 , 70 and  7 
0
§3 Negative Exponents
Finally, let’s look at something like
x3
. We know that by the quotient rule, we can simplify this to x 5 . How else
8
x
x3
xxx
1
1

 5 . Hence 5  x 5 . Another way to think about
8
x
xxxxxxxx x
x
4
4
negative exponents is as follows: try to simplify x  x . We see that by the product rule, the answer is x 0  1!
Hence the two terms are reciprocals of each other! That is exactly what a negative exponent is. It is the
reciprocal of the same base but with a negative exponent!
can we interpret this? Well,
Generally, a  n 
1
. Hence if you end up with an expression that has a negative exponent, we can ‘make’ this
an
2
2
4
1
3
2
and       .
5
3
2
3 9
Make sure whenever you change the base to its reciprocal we also change the sign of the exponent. That means
term have a positive exponent simply by taking the reciprocal of the base! So 35 
4
4
2 5
something like      . It goes both ways. However, we ALWAYS LEAVE ANSWERS WITH POSITIVE
5 2
EXPONENTS!!!
Generally,
a m bn
a
 m and  
n
b
a
b
m
b
 
a
m
PRACTICE
3
2) Simplify 43 ,  
4
2
and x 5
3) Simplify and leave final answer with positive exponents only: x3 y 6
4) Simplify and leave final answer with positive exponents only:
2 4
33
5) Simplify and leave final answer with positive exponents only:
3x 4 y 5
3x6 y
x y 
6) Simplify and leave final answer with positive exponents only:
 x y
3
4
5 2
2