How to Multiply and Divide Monomials? (+FREE Worksheet!)

Multiplying and dividing monomials both rely on the same core exponent rules, and once you know them you can handle any problem in seconds. Multiplication uses the product rule (add exponents), while division uses the quotient rule (subtract exponents). This page gives you both rules, step-by-step examples, two video lessons, and plenty of practice problems.

What Are Monomials?

A monomial is a single algebraic term, a number, a variable, or a product of numbers and variables with non-negative integer exponents. Examples: \(\color{blue}{5}\), \(\color{blue}{3x}\), \(\color{blue}{-2x^{3}}\), \(\color{blue}{7x^{2}y}\). There are no plus or minus signs inside a monomial.

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Multiplying Monomials: The Product Rule

Rule: Multiply the coefficients and add the exponents of matching variables.

x^a ⋅ x^b = x^{\(\color{blue}{a + b}\)}

Multiplication Examples

• \(\color{blue}{(6x^{3})(2x^{2}) = 12x^{5}}\)  —  \(\color{blue}{6 \times 2 = 12}\); \(\color{blue}{3 + 2 = 5}\)
• (4x²y³)(3xy) = 12x³y^4  —  coefficients: \(\color{blue}{4 \times 3 = 12}\); x: \(\color{blue}{2+1=3}\); y: \(\color{blue}{3+1=4}\)

Dividing Monomials: The Quotient Rule

Rule: Divide the coefficients and subtract the exponents of matching variables.

x^a ÷ x^b = x^{\(\color{blue}{a – b}\)}  (x ≠ 0)

Division Examples

• \(\color{blue}{12x^{7} \div 4x^{3} = 3x^{4}}\)  —  \(\color{blue}{12 \div 4 = 3}\); \(\color{blue}{7 – 3 = 4}\)
• \(\color{blue}{20a^{5} \div 5a^{2} = 4a^{3}}\)  —  \(\color{blue}{20 \div 5 = 4}\); \(\color{blue}{5 – 2 = 3}\)
• \(\color{blue}{(-15m^{6}) \div (3m^{2}) = -5m^{4}}\)

Step-by-Step Summary

1. Identify the coefficients and the variables in each monomial.
2. Multiplication: multiply coefficients, add exponents.
3. Division: divide coefficients, subtract exponents (numerator minus denominator).
4. Handle each variable independently when more than one variable is present.
5. Simplify the coefficient and write in standard form.

Watch: Multiplying Monomials (Video Lesson)

This Khan Academy lesson covers the product rule with clear, worked problems:

Multiplying and Dividing Monomials — Worked Examples

Example 1: Multiply \(\color{blue}{(6x^{3})(2x^{2})}\).

Coefficients: \(\color{blue}{6 \times 2 = 12}\). Exponents: \(\color{blue}{3 + 2 = 5}\).
Answer: \(\color{blue}{12x^{5}}\)

Example 2: Divide \(\color{blue}{12x^{7} \div 4x^{3}}\).

Coefficients: \(\color{blue}{12 \div 4 = 3}\). Exponents: \(\color{blue}{7 – 3 = 4}\).
Answer: \(\color{blue}{3x^{4}}\)

Example 3: Divide \(\color{blue}{20a^{5} \div 5a^{2}}\).

Coefficients: \(\color{blue}{20 \div 5 = 4}\). Exponents: \(\color{blue}{5 – 2 = 3}\).
Answer: \(\color{blue}{4a^{3}}\)

Example 4: Divide \(\color{blue}{(-15m^{6}) \div (3m^{2})}\).

Coefficients: \(\color{blue}{-15 \div 3 = -5}\). Exponents: \(\color{blue}{6 – 2 = 4}\).
Answer: \(\color{blue}{-5m^{4}}\)

Watch: Dividing Monomials (Video Lesson)

This video focuses on the quotient rule and shows how to divide monomials step by step:

Exercises for Multiplying and Dividing Monomials

Multiply or divide as indicated.

1. \(\color{blue}{(5x^{4})(3x^{2})}\)
2. \(\color{blue}{24y^{8} \div 6y^{3}}\)
3. \(\color{blue}{(-4a^{3})(2a^{5})}\)
4. \(\color{blue}{18m^{7} \div (-3m^{4})}\)
5. (2x²y)(5xy³)
6. \(\color{blue}{-36a^{6}b^{4} \div 9a^{2}b}\)

Answers

1. \(\color{blue}{15x^{6}}\)
2. \(\color{blue}{4y^{5}}\)
3. \(\color{blue}{-8a^{8}}\)
4. \(\color{blue}{-6m^{3}}\)
5. \(\color{blue}{10x^{3}y^{4}}\)
6. \(\color{blue}{-4a^{4}b^{3}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Multiplying and Dividing Monomials just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work. They’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Properties of Exponents Worksheet
• Download Multiplying Polynomials Worksheet

Frequently Asked Questions

How do you multiply and divide monomials?

To multiply, multiply the coefficients and add the exponents of the same variable. To divide, divide the coefficients and subtract the exponents. For example, (3x²)(4x⁵) = 12x⁷, and 20x⁶ ÷ 5x² = 4x⁴. Work one variable at a time when there are several, and remember that a variable with no exponent shown has an exponent of 1.

How do you multiply monomials and polynomials?

Distribute the monomial to each term of the polynomial, using the product rule on every piece. For 2a(3a² − a + 4), multiply 2a × 3a² = 6a³, 2a × (−a) = −2a², and 2a × 4 = 8a. The answer is 6a³ − 2a² + 8a, with the same number of terms as the polynomial you started with.

How do you divide monomials and polynomials?

Divide each term of the polynomial by the monomial, then simplify each piece with the quotient rule. For (12x³ − 8x² + 4x) ÷ 4x, you get 12x³ ÷ 4x = 3x², −8x² ÷ 4x = −2x, and 4x ÷ 4x = 1, so the result is 3x² − 2x + 1. Don’t drop that last 1.

How do you multiply and divide monomials with negative exponents?

Use the same rules, add exponents when multiplying and subtract when dividing, then rewrite any negative exponent as a fraction. For example, x³ · x⁻⁵ = x⁻² = 1/x². In division, 6y² ÷ 3y⁵ = 2y⁻³ = 2/y³. Final answers are usually written with positive exponents only.

How do you add, subtract, multiply and divide monomials?

Adding and subtracting work only with like terms, and the exponents stay the same: 3x² + 5x² = 8x². Multiplying and dividing work with any monomials and change the exponents: 3x² × 5x² = 15x⁴ and 15x⁴ ÷ 3x = 5x³. Mixing up these two sets of rules, like writing 3x² + 5x² = 8x⁴, is a very common error.

Related Topics

• Multiplying Monomials
• Multiplying a Polynomial and a Monomial
• Simplifying Polynomials
• Multiplying Binomials
• Factoring Polynomials