Objective Learn the notion of dividing
monomials, and the idea of zero and negative numbers as
It is a good idea that you experiment with computing quotients
of powers of 2 in order to see if you can discover the formula
for division of powers by yourself.
Dividing Powers of the Same Number
To understand division of powers, look at the table that
appears below. It shows the results when 64, which equals 2
6 , is divided by various powers of 2.
Do you see any pattern in these numbers? The result is found
by subtracting the exponents.
Divide x 4 by x 2 .
Expand x 4 into x Â· x Â· x Â· x and x 2
into x Â· x . Then use these expressions to write a fraction.
Now cancel two of the x s from both numerator and
It is very important that you understand that
cancellation just means recognizing that
You should understand that canceling is just shorthand for a
process involving the properties of fractions. Any number raised
to the power 1 is that number itself.
Quotient of Powers
If we divide a power of a number or variable by another
(smaller) power of the same number or variable, the result is the
original number raised to the power given by the difference of
the two original powers.
Write the quotient and expand the numerator and the
denominator into products of xs.
This shows that the formula for the quotient of powers is
This formula can be used to simplify any quotient of
Negative Numbers and Zero in the Exponent
We have only defined exponentiation by positive integers, not
for zero or for negative numbers. Remind students that when m
> n ,
If we let m < n , then we get
On the other hand, , since any number divided by itself is 1.
This leads to the following definition.
a 0 is defined to be equal to 1.
What would the formula suggest if m < n? For example, let m =
1 and n = 2.
On the other hand,
So, the formula suggests that a -1 should be
defined to be . More generally, it
suggests that a -n should be defined to be .
a -n is defined to be equal to .
At this point, you may be confused by the way the formula was
used to generate a definition for negative powers. It was used
only as a guide to suggest what a definition should be, but that
once we make the definition, we can work with these exponents in
exactly the same way that we did with positive exponents. With
this definition, these exponents satisfy all the laws of
exponents that you should already know. The definition of
negative exponents can be used to simplify quotients of