# Polynomials

## Divide Monomials


### Simplify Expressions Using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.
As before, we'll try to discover a property by looking at some examples.

Notice that in each case the bases were the same and we subtracted the exponents.
1. When the larger exponent was in the numerator, we were left with factors in the numerator and  in the denominator, which we simplified.
2. When the larger exponent was in the denominator, we were left with factors in the denominator, and  in the numerator, which could not be simplified.
We write:
A couple of examples with numbers may help to verify this property.
When we work with numbers and the exponent is less than or equal to  we will apply the exponent. When the exponent is greater than , we leave the answer in exponential form.

### Simplify Expressions with Zero Exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like  From earlier work with fractions, we know that
In words, a number divided by itself is  So  for any  (), since any number divided by itself is 
The Quotient Property of Exponents shows us how to simplify  when  and when  by subtracting exponents. What if ?
Now we will simplify  in two ways to lead us to the definition of the zero exponent.
Consider first  which we know is 

We see  simplifies to a  and to . So .
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  We can use the product to a power rule to rewrite this expression.

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

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

Now we will look at an example that will lead us to the Quotient to a Power Property.
Notice that the exponent applies to both the numerator and the denominator.
We see that  is 

This leads to the Quotient to a Power Property for Exponents.

An example with numbers may help you understand this property:

### Simplify Expressions by Applying Several Properties

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.

### Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.
When we divide monomials with more than one variable, we write one fraction for each variable.
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.

### Key Concepts

1. Equivalent Fractions Property

2. Zero Exponent


3. Quotient Property for Exponents

4. Quotient to a Power Property for Exponents




### Practice Makes Perfect

Simplify Expressions Using the Quotient Property of 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.

### Mixed Practice


### Everyday Math


### Writing Exercises


### Self Check

ⓐ 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?
