6.5 Divide Monomials

Learning Objectives

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

Simplify Expressions Using the Quotient Property for Exponents

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.

Simplify Expressions with an Exponent of Zero

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:

This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.

We see amam simplifies to a0 and to 1. So a0=1.

Zero Exponent

If a is a non-zero number, then a0=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 (2x)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.

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.

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

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.

Media

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

  • Rational Expressions
  • Dividing Monomials
  • Dividing Monomials 2

Section 6.5 Exercises

Practice Makes Perfect

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.

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.This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “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,” and “divide monomials.” The rest of the cells are blank.

ⓑ 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?