By the end of this section, you will be able to:

- Simplify expressions using the Quotient Property for Exponents
- Simplify expressions with zero exponents
- Simplify expressions using the quotient to a Power Property
- Simplify expressions by applying several properties
- Divide monomials

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

As before, we’ll try to discover a property by looking at some examples.

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

This leads to the *Quotient Property for Exponents*.

A couple of examples with numbers may help to verify this property.

Notice the difference in the two previous examples:

- If we start with more factors in the numerator, we will end up with factors in the numerator.
- If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like *amam*. From your earlier work with fractions, you know that:

Now we will simplify *amam* in two ways to lead us to the definition of the zero exponent. In general, for *a*≠0:

We see *amam* simplifies to *a*0 and to 1. So *a*^{0}=1.

If *a* is a non-zero number, then *a*^{0}=1.

Any nonzero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let’s look at (2*x*)0. We can use the product to a power rule to rewrite this expression.

This tells us that any nonzero expression raised to the zero power is one.

Now we will look at an example that will lead us to the Quotient to a Power Property.

An example with numbers may help you understand this property:

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

Access these online resources for additional instruction and practice with dividing monomials:

- Rational Expressions
- Dividing Monomials
- Dividing Monomials 2

**Simplify Expressions Using the Quotient Property for Exponents**

In the following exercises, simplify.

**Simplify Expressions with Zero Exponents**

In the following exercises, simplify.

**Simplify Expressions Using the Quotient to a Power Property**

In the following exercises, simplify.

**Simplify Expressions by Applying Several Properties**

In the following exercises, simplify.

**Divide Monomials**

In the following exercises, divide the monomials.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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