Learn on PengiOpenstax Prealgebre 2EChapter 10: Polynomials

Lesson 4: Divide Monomials

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn to divide monomials by applying the Quotient Property of Exponents, the zero exponent rule, and the Quotient to a Power Property. Learners practice simplifying algebraic expressions by subtracting exponents when dividing terms with the same base, including cases where the larger exponent appears in the numerator or denominator. The lesson builds on multiplication properties of exponents to develop a complete toolkit for working with monomial expressions.

Section 1

📘 Divide Monomials

New Concept

In this lesson, we'll master dividing monomials. You'll learn how to use the Quotient Property of Exponents by subtracting exponents, explore the meaning of a zero exponent ( $a^0=1$ ), and apply these new skills to simplify algebraic fractions.

What’s next

Now that you have the core idea, you'll tackle worked examples on simplifying quotients and then test your skills with a series of practice cards.

Section 2

Quotient property of exponents

Property

If $a$ is a real number, $a \neq 0$ , and $m$ , $n$ are whole numbers, then

\frac{a^m}{a^n} = a^{m-n}, \quad m > n \text{ and } \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

Examples

To simplify $\frac{x^9}{x^4}$ , you subtract the exponents since $9 > 4$ . The result is $x^{9-4} = x^5$ .

To simplify $\frac{c^5}{c^8}$ , the larger exponent is in the denominator, so the result is $\frac{1}{c^{8-5}} = \frac{1}{c^3}$ .

Section 3

Zero exponent

Property

If $a$ is a non-zero number, then $a^0 = 1$ .
Any nonzero number raised to the zero power is 1.

Examples

Any non-zero number to the zero power is $1$ . For example, $150^0 = 1$ .

For the expression $(5x)^0$ , the entire quantity is raised to the zero power, so the result is $1$ .

Section 4

Quotient to a power property

Property

If $a$ and $b$ are real numbers, $b \neq 0$ , and $m$ is a counting number, then

\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}

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

Examples

To simplify $\left(\frac{2}{3}\right)^4$ , you apply the exponent to both the numerator and the denominator: $\frac{2^4}{3^4} = \frac{16}{81}$ .

To simplify $\left(\frac{p}{q}\right)^7$ , you raise both the numerator and denominator to the 7th power, resulting in $\frac{p^7}{q^7}$ .

Section 5

Summary of exponent properties

Property

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

Property	Formula
Product Property	$a^m \cdot a^n = a^{m+n}$
Power Property	$(a^m)^n = a^{mn}$
Product to a Power Property	$(ab)^m = a^m b^m$
Quotient Property	$\frac{a^m}{a^n} = a^{m-n}, a \neq 0, m > n$
$\frac{a^m}{a^n} = \frac{1}{a^{n-m}}, a \neq 0, n > m$
Zero Exponent Definition	$a^0 = 1, a \neq 0$
Quotient to a Power Property	$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, b \neq 0$

Examples

To simplify $\frac{(x^4)^5}{x^{12}}$ , first use the Power Property on the numerator to get $\frac{x^{20}}{x^{12}}$ , then use the Quotient Property to get $x^8$ .

Section 6

Divide monomials

Property

To divide monomials, rewrite the division as a fraction. Separate the expression into numerical coefficients and each variable. Simplify the coefficients and use the Quotient Property of Exponents for each variable.

Examples

To find the quotient $72x^8 \div 9x^3$ , rewrite it as $\frac{72x^8}{9x^3}$ . Simplify the numbers and subtract the exponents: $8x^{8-3} = 8x^5$ .

To divide $\frac{45a^6b^4}{5ab^2}$ , handle each part separately: $\frac{45}{5} \cdot \frac{a^6}{a} \cdot \frac{b^4}{b^2}$ . This simplifies to $9 \cdot a^5 \cdot b^2 = 9a^5b^2$ .

Book overview

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Chapter 10: Polynomials

Back to Openstax Prealgebre 2E

Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Divide Monomials
Learn Practice
Lesson 5
Lesson 5: Integer Exponents and Scientific Notation
Learn Practice
Lesson 6
Lesson 6: Introduction to Factoring Polynomials
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Divide Monomials

New Concept

What’s next

Now that you have the core idea, you'll tackle worked examples on simplifying quotients and then test your skills with a series of practice cards.

Section 2

Quotient property of exponents

Property

If $a$ is a real number, $a \neq 0$ , and $m$ , $n$ are whole numbers, then

\frac{a^m}{a^n} = a^{m-n}, \quad m > n \text{ and } \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

Examples

To simplify $\frac{x^9}{x^4}$ , you subtract the exponents since $9 > 4$ . The result is $x^{9-4} = x^5$ .

To simplify $\frac{c^5}{c^8}$ , the larger exponent is in the denominator, so the result is $\frac{1}{c^{8-5}} = \frac{1}{c^3}$ .

Section 3

Zero exponent

Property

If $a$ is a non-zero number, then $a^0 = 1$ .
Any nonzero number raised to the zero power is 1.

Examples

Any non-zero number to the zero power is $1$ . For example, $150^0 = 1$ .

For the expression $(5x)^0$ , the entire quantity is raised to the zero power, so the result is $1$ .

Section 4

Quotient to a power property

Property

If $a$ and $b$ are real numbers, $b \neq 0$ , and $m$ is a counting number, then

\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}

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

Examples

To simplify $\left(\frac{2}{3}\right)^4$ , you apply the exponent to both the numerator and the denominator: $\frac{2^4}{3^4} = \frac{16}{81}$ .

To simplify $\left(\frac{p}{q}\right)^7$ , you raise both the numerator and denominator to the 7th power, resulting in $\frac{p^7}{q^7}$ .

Section 5

Summary of exponent properties

Property

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

Property	Formula
Product Property	$a^m \cdot a^n = a^{m+n}$
Power Property	$(a^m)^n = a^{mn}$
Product to a Power Property	$(ab)^m = a^m b^m$
Quotient Property	$\frac{a^m}{a^n} = a^{m-n}, a \neq 0, m > n$
$\frac{a^m}{a^n} = \frac{1}{a^{n-m}}, a \neq 0, n > m$
Zero Exponent Definition	$a^0 = 1, a \neq 0$
Quotient to a Power Property	$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, b \neq 0$

Examples

To simplify $\frac{(x^4)^5}{x^{12}}$ , first use the Power Property on the numerator to get $\frac{x^{20}}{x^{12}}$ , then use the Quotient Property to get $x^8$ .

Section 6

Divide monomials

Property

Examples

To find the quotient $72x^8 \div 9x^3$ , rewrite it as $\frac{72x^8}{9x^3}$ . Simplify the numbers and subtract the exponents: $8x^{8-3} = 8x^5$ .

To divide $\frac{45a^6b^4}{5ab^2}$ , handle each part separately: $\frac{45}{5} \cdot \frac{a^6}{a} \cdot \frac{b^4}{b^2}$ . This simplifies to $9 \cdot a^5 \cdot b^2 = 9a^5b^2$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 10: Polynomials

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Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Divide Monomials
Learn Practice
Lesson 5
Lesson 5: Integer Exponents and Scientific Notation
Learn Practice
Lesson 6
Lesson 6: Introduction to Factoring Polynomials
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

📘 Divide Monomials

New Concept

What’s next

Now that you have the core idea, you'll tackle worked examples on simplifying quotients and then test your skills with a series of practice cards.

Section 2

Quotient property of exponents

Property

If $a$ is a real number, $a \neq 0$ , and $m$ , $n$ are whole numbers, then

\frac{a^m}{a^n} = a^{m-n}, \quad m > n \text{ and } \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

Examples

To simplify $\frac{x^9}{x^4}$ , you subtract the exponents since $9 > 4$ . The result is $x^{9-4} = x^5$ .

To simplify $\frac{c^5}{c^8}$ , the larger exponent is in the denominator, so the result is $\frac{1}{c^{8-5}} = \frac{1}{c^3}$ .

Section 3

Zero exponent

Property

If $a$ is a non-zero number, then $a^0 = 1$ .
Any nonzero number raised to the zero power is 1.

Examples

Any non-zero number to the zero power is $1$ . For example, $150^0 = 1$ .

For the expression $(5x)^0$ , the entire quantity is raised to the zero power, so the result is $1$ .

Section 4

Quotient to a power property

Property

If $a$ and $b$ are real numbers, $b \neq 0$ , and $m$ is a counting number, then

\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}

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

Examples

To simplify $\left(\frac{2}{3}\right)^4$ , you apply the exponent to both the numerator and the denominator: $\frac{2^4}{3^4} = \frac{16}{81}$ .

To simplify $\left(\frac{p}{q}\right)^7$ , you raise both the numerator and denominator to the 7th power, resulting in $\frac{p^7}{q^7}$ .

Section 5

Summary of exponent properties

Property

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

Property	Formula
Product Property	$a^m \cdot a^n = a^{m+n}$
Power Property	$(a^m)^n = a^{mn}$
Product to a Power Property	$(ab)^m = a^m b^m$
Quotient Property	$\frac{a^m}{a^n} = a^{m-n}, a \neq 0, m > n$
$\frac{a^m}{a^n} = \frac{1}{a^{n-m}}, a \neq 0, n > m$
Zero Exponent Definition	$a^0 = 1, a \neq 0$
Quotient to a Power Property	$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, b \neq 0$

Examples

To simplify $\frac{(x^4)^5}{x^{12}}$ , first use the Power Property on the numerator to get $\frac{x^{20}}{x^{12}}$ , then use the Quotient Property to get $x^8$ .

Section 6

Divide monomials

Property

Examples

To find the quotient $72x^8 \div 9x^3$ , rewrite it as $\frac{72x^8}{9x^3}$ . Simplify the numbers and subtract the exponents: $8x^{8-3} = 8x^5$ .

To divide $\frac{45a^6b^4}{5ab^2}$ , handle each part separately: $\frac{45}{5} \cdot \frac{a^6}{a} \cdot \frac{b^4}{b^2}$ . This simplifies to $9 \cdot a^5 \cdot b^2 = 9a^5b^2$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 10: Polynomials

Back to Openstax Prealgebre 2E

Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Divide Monomials
Learn Practice
Lesson 5
Lesson 5: Integer Exponents and Scientific Notation
Learn Practice
Lesson 6
Lesson 6: Introduction to Factoring Polynomials
Learn Practice

Lesson 4: Divide Monomials

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 10: Polynomials

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Divide Monomials

Lesson 5: Integer Exponents and Scientific Notation

Lesson 6: Introduction to Factoring Polynomials

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 10: Polynomials

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Divide Monomials

Lesson 5: Integer Exponents and Scientific Notation

Lesson 6: Introduction to Factoring Polynomials

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Divide Monomials

Lesson 5: Integer Exponents and Scientific Notation

Lesson 6: Introduction to Factoring Polynomials