There are a few special exponent properties that deal with exponents that are not positive. The first is considered in the following example, which is worded out 2 different ways:

Use the quotient rule to subtract exponents  

Our Solution, but now we consider the problem the second way:  

Rewrite exponents as repeated multiplication  

Reduce out all the  

Our Solution, when we combine the two solutions, we get:  

Our final result 
This final result is an important property known as the zero property of exponents:

Zero power rule  

Our Solution 
Another property we will consider here deals with negative exponents. Again we will solve the following example two ways.

Using the quotient rule, subtract exponents  

Our Solution, but we will also solve this problem another way  

Rewrite exponents as repeated multiplication  

Reduce three out of top and bottom  

Simplify to exponents  

Our Solution, putting these solutions together gives:  

Our Final Solution 
This example illustrates an important property of exponents. Negative exponents yield the reciprocal of the base. Once we take the reciprocal the exponent is now positive. Also, it is important to note a negative exponent does not mean the expression is negative, only that we need the reciprocal of the base. Following are the properties of negative exponents.

Negative exponents on b, d, and e need to flip  

Our Solution 
As we simplified our fraction we took special care to move the bases that had a negative exponent, but the expression itself did not become negative because of those exponents. Also, it is important to remember that exponents only affect what they are attached to. The 2 in the denominator of the above example does not have an exponent on it, so it does not move with the d.
Simplifying with negative exponents is much the same as simplifying with positive exponents. It is the advice of the author to keep the negative exponents until the end of the problem and then move them around to their correct location (numerator or denominator). As we do this it is important to be very careful of rules for adding, subtracting, and multiplying with negatives. This is illustrated in the following examples:

Simplify numerator with product rule, adding exponents  

Use Quotient rule to subtract exponents, be careful with the negatives!  

Our Solution 
EXAMPLE

Simplify numerator with product rule, adding exponents  

Quotient rule to subtract exponents, be careful with negatives!  







Negative exponent needs to move down to denominator  

Our Solution 
EXAMPLE

In numerator, use power rule with , multiplying exponents. In denominator,  

In numerator, use product rule to add exponents  

Use quotient rule to subtract exponents, be careful with negatives  




Move and b to denominator because of negative exponents  

Evaluate  

Our Solution 