How to Multiply Monomials? (+FREE Worksheet!)

Multiplying monomials is a core Algebra 1 skill that relies on two simple rules: multiply the coefficients together and add the exponents of matching variables. Whether you are multiplying single-variable or multi-variable monomials, the same rules always apply. This page covers everything you need, concept, rules, worked examples, videos, and practice.

What Is a Monomial?

A monomial is a single-term algebraic expression consisting of a number, a variable, or a product of numbers and variables with non-negative integer exponents. Examples: \(\color{blue}{7}\), \(\color{blue}{4x}\), \(\color{blue}{-3x^{2}}\), \(\color{blue}{5x^{2}y}\). It has no addition or subtraction.

Algebra I for Beginners 2027 The Ultimate Step by Step Guide to Acing Algebra I

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rules for Multiplying Monomials

1. Multiply the Coefficients

Multiply the numerical parts (coefficients) like ordinary multiplication, keeping track of positive and negative signs.

• \(\color{blue}{3 \times 4 = 12}\)
• \(\color{blue}{(-2) \times 5 = -10}\)

2. Apply the Product Rule for Exponents

For the variable parts, use the Product Rule: x^a ⋅ x^b = x^{\(\color{blue}{a + b}\)}. When the bases are the same, add the exponents.

• \(\color{blue}{x^{2} \cdot x^{3} = x^{5}}\)
• \(\color{blue}{a^{3} \cdot a^{4} = a^{7}}\)

3. Handle Multiple Variables Separately

For monomials with more than one variable, apply the product rule to each variable independently.

• (4x²y)(3xy³) = (4 · 3)(x²+1)(y¹+3) = 12x³y^4

Step-by-Step Summary

1. Multiply the numerical coefficients (watch the signs).
2. For each variable, add the exponents from both monomials.
3. Write the result: coefficient first, then variables in alphabetical order.

Watch: Multiplying Monomials (Concept Lesson)

Khan Academy explains the product rule and works through examples at a steady pace:

Multiplying Monomials — Worked Examples

Example 1: Multiply \(\color{blue}{(3x^{2})(4x^{3})}\).

Multiply coefficients: \(\color{blue}{3 \times 4 = 12}\). Add exponents: \(\color{blue}{x^{2+3} = x^{5}}\).
Answer: \(\color{blue}{12x^{5}}\)

Example 2: Multiply \(\color{blue}{(-2a^{3})(5a^{4})}\).

Coefficients: \(\color{blue}{(-2)(5) = -10}\). Variable: \(\color{blue}{a^{3+4} = a^{7}}\).
Answer: \(\color{blue}{-10a^{7}}\)

Example 3: Multiply (4x²y)(3xy³).

Coefficients: \(\color{blue}{4 \times 3 = 12}\). For x: \(\color{blue}{x^{2+1} = x^{3}}\). For y: \(\color{blue}{y^{1+3} = y^{4}}\).
Answer: \(\color{blue}{12x^{3}y^{4}}\)

Example 4: Multiply (-3m²n)(-2mn²).

Coefficients: \(\color{blue}{(-3)(-2) = 6}\). For m: \(\color{blue}{m^{2+1} = m^{3}}\). For n: \(\color{blue}{n^{1+2} = n^{3}}\).
Answer: \(\color{blue}{6m^{3}n^{3}}\)

More Practice: Step-by-Step Video

This Khan Academy example video focuses on single-variable monomials with clear step-by-step explanation:

Exercises for Multiplying Monomials

Find each product.

1. \(\color{blue}{(2x^{3})(5x^{2})}\)
2. \(\color{blue}{(-3y)(4y^{4})}\)
3. (2a²b)(3ab²)
4. (-5xy²)(-2x²y)
5. (4m²n³)(-3mn)

Answers

1. \(\color{blue}{10x^{5}}\)
2. \(\color{blue}{-12y^{5}}\)
3. \(\color{blue}{6a^{3}b^{3}}\)
4. \(\color{blue}{10x^{3}y^{3}}\)
5. \(\color{blue}{-12m^{3}n^{4}}\)

The Ultimate Algebra Bundle 2027 From Pre-Algebra to Algebra II

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Free Multiplying Monomials Worksheet

Ready to practice on your own? Download our free Multiplying Monomials worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Multiplying Monomials before a quiz or test.

Download Multiplying Polynomials Worksheet

Frequently Asked Questions

What is the rule for multiplying monomials?

Multiply the coefficients, then multiply the variables by adding the exponents of each matching base. For (−2x³)(5x⁴), multiply −2 × 5 = −10 and x³ · x⁴ = x⁷, so the product is −10x⁷. Keep track of the sign of the coefficient first; it is the easiest part to lose.

What is the product rule for multiplying monomials?

The product rule says xᵃ · xᵇ = xᵃ⁺ᵇ when the bases are the same. You can see why by writing it out: x² · x³ = (x · x)(x · x · x) = x⁵, five x’s in all. The rule only applies to matching bases, so x² · y³ stays x²y³.

How do you multiply monomials with different variables?

Multiply the coefficients, then handle each variable separately, adding exponents only for matching letters. For (3a²b)(4ab³), 3 × 4 = 12, a² · a = a³, and b · b³ = b⁴, giving 12a³b⁴. A variable that appears in only one monomial, like c in (2c)(5d), simply carries over: 10cd.

How do you multiply monomials with fractions?

Multiply the fraction coefficients like ordinary fractions, then add the exponents. For (1/2 x²)(2/3 x³), 1/2 × 2/3 = 2/6 = 1/3, so the product is (1/3)x⁵. Simplifying before multiplying can save work: (3/4 y)(8y²) = 6y³, because 8 ÷ 4 = 2 and 3 × 2 = 6.

How do you multiply 3 monomials?

Multiply them two at a time, or multiply all the coefficients together and then add all the exponents of each variable. For (2x)(3x²)(−x⁴), the coefficients give 2 × 3 × (−1) = −6 and the exponents give 1 + 2 + 4 = 7, so the answer is −6x⁷. Because multiplication is commutative, the order you choose does not matter.

How do you multiply monomials with exponents?

Add exponents when you multiply powers of the same base, and multiply exponents when a whole monomial is raised to a power. So 4x² · 3x⁶ = 12x⁸, while (2x³)² = 2² · x⁶ = 4x⁶. A common slip is writing (2x³)² = 2x⁶; the coefficient must be squared too.

Related Topics

• Multiplying and Dividing Monomials
• Multiplying a Polynomial and a Monomial
• Simplifying Polynomials
• Multiplying Binomials
• Adding and Subtracting Polynomials