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.

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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. \(\color{blue}{-12m^{3}n^{4}}\)

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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 product rule for exponents?

The product rule states that when you multiply two powers with the same base, you add their exponents: \(\color{blue}{x^{a} \cdot x^{b} = x^{a+b}}\). For example, \(\color{blue}{x^{3} \cdot x^{2} = x^{5}}\).

What happens to the sign when multiplying negative monomials?

Negative × negative = positive; negative × positive = negative. Apply the same sign rules you use for regular integers: \(\color{blue}{(-4x)(3x) = -12x^{2}}\), and \(\color{blue}{(-4x)(-3x) = 12x^{2}}\).

Can you multiply monomials with different variables?

Yes. Different variables cannot have their exponents combined. Write them side by side: (3x)(2y) = 6xy — the x and y remain separate.

Related Topics

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