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Mathematics LibreTexts

5.1: Simplify Monomials

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    Jan 4, 2021
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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
Note

Before you get started, take this readiness quiz.

  1. Simplify: .
    If you missed this problem, review Example 1.6.4.
  2. Simplify: .
    If you missed this problem, review Example 6.2.22.
  3. Simplify:
    If you missed this problem, review Example 1.6.10.

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.

SUMMARY OF EXPONENT PROPERTIES FOR MULTIPLICATION

If a and b are real numbers, and m and n are whole numbers, then

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.

EQUIVALENT FRACTIONS PROPERTY

If a, b, and c are whole numbers where .

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.

QUOTIENT PROPERTY FOR EXPONENTS

If a is a real number, , and m and n are whole numbers, then

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

Example

Simplify:

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

1.

Since 9 > 7, there are more factors of x in the numerator.
Use the Quotient Property,
Simplify.

2.

Since 10 > 2, there are more factors of x in the numerator.
Use the Quotient Property,
Simplify.
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Try It

Simplify:

Answer
Try It

Simplify:

Answer
Example

Simplify:

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

1.

Since 12 > 8, there are more factors of b in the denominator.
Use the Quotient Property,
Simplify.

2.

Since 5 > 3, there are more factors of 3 in the denominator.
Use the Quotient Property,
Simplify.
Simplify.
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Try It

Simplify:

Answer
Try It

Simplify:

Answer

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.

Example

Simplify:

Solution

1. Is the exponent of a larger in the numerator or denominator? Since 9 > 5, there are more a's in the denominator and so we will end up with factors in the denominator.

 
Use the Quotient Property,
Simplify.

2. Notice there are more factors of xx in the numerator, since 11 > 7. So we will end up with factors in the numerator.

 
Use the Quotient Property,
Simplify.
Try It

Simplify:

Answer
Try It

Simplify:

Answer

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 . From your earlier work with fractions, you know that:

In words, a number divided by itself is 1. So, , for any , since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify when and when by subtracting exponents. What if ?

Consider , which we know is 1.

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

We see simplifies to and to 1. So .

ZERO EXPONENT

If a is a non-zero number, then .

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.

Example

Simplify:

Solution

The definition says any non-zero number raised to the zero power is 1.

Try It

Simplify:

Answer
  1. 1
  2. 1
Try It

Simplify:

Answer
  1. 1
  2. 1

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 nonzero expression raised to the zero power is one.

Example

Simplify:

  2. .

Solution

Try It

Simplify:

  2. .
Answer
  1. 1
  2. 1
Try It

Simplify:

  2. .
Answer
  1. 1
  2. 1

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.

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

QUOTIENT TO A POWER PROPERTY FOR EXPONENTS

If a and b are real numbers, , and m is a counting number, then

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

Example

Simplify:

Solution

1.

 
Use the Quotient Property,
Simplify.

2.

 
Use the Quotient Property,
Simplify.

3.

 
Raise the numerator and denominator to the third power.
Try It

Simplify:

Answer
Try It

Simplify:

Answer

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.

SUMMARY OF EXPONENT PROPERTIES

If a and b are real numbers, and m and n are whole numbers, then

Example

Simplify:

Solution

Try It

Simplify:

Answer

Try It

Simplify:

Answer

Example

Simplify:

Solution

Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.

Try It

Simplify .

Answer

1

Try It

Simplify .

Answer

1

Example

Simplify:

Solution

Try It

Simplify:

Answer

Try It

Simplify:

Answer

Example

Simplify:

Solution

Try It

Simplify:

Answer

Try It

Simplify:

Answer

Example

Simplify:

Solution

Try It

Simplify:

Answer

Try It

Simplify:

Answer

Example

Simplify:

Solution

Try It

Simplify:

Answer

Try It

Simplify:

Answer

Example

Simplify:

Solution

Try It

Simplify:

Answer

9

Try It

Simplify:

Answer

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.

Example

Find the quotient:

Solution

Try It

Find the quotient:

Answer

Try It

Find the quotient:

Answer

Example

Find the quotient:

Solution

When we divide monomials with more than one variable, we write one fraction for each variable.

Try It

Find the quotient:

Answer

Try It

Find the quotient:

Answer

Example

Find the quotient:

Solution

Try It

Find the quotient:

Answer

Try It

Find the quotient:

Answer

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.

Example

Find the quotient:

Solution

Be very careful to simplify by dividing out a common factor, and to simplify the variables by subtracting their exponents.

Try It

Find the quotient:

Answer

Try It

Find the quotient:

Answer

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.

Example

Find the quotient:

Solution

Answer

Try It

Find the quotient:

Answer

Try It

Find the quotient:

Answer

Note

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

Key Concepts

  • Quotient Property for Exponents:
    • If a is a real number, , and m,n are whole numbers, then:
  • Zero Exponent
    • If a is a non-zero number, then .
  • Quotient to a Power Property for Exponents:
    • If a and b are real numbers, , and mm is a counting number, then:
    • To raise a fraction to a power, raise the numerator and denominator to that power.
  • Summary of Exponent Properties
    • If a,b are real numbers and m,nm,n are whole numbers, then

5.1: Simplify Monomials is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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