How to Multiply Rational Expressions? (+FREE Worksheet!)

Multiplying rational expressions is one of the most satisfying operations in algebra because it rewards you for factoring: factor the numerators and denominators, cancel any common factors across the multiplication, and then multiply what remains. The process is exactly like multiplying ordinary fractions, with the bonus step of factoring and simplifying first. Mastering multiplying rational expressions is essential for Algebra 1 and all subsequent math courses.

What Does It Mean to Multiply Rational Expressions?

To multiply two rational expressions, multiply their numerators together and their denominators together:

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\(\color{blue}{(\frac{A}{B}) \times (\frac{C}{D}) = \frac{(A \cdot C)}{(B \cdot D)}}\)

The key is to factor first, then cancel before multiplying, so the final answer is already in simplest form. Also note any domain restrictions from the original denominators.

How to Multiply Rational Expressions

Step 1 — Factor all numerators and denominators

Factor every polynomial completely: difference of squares, trinomials, GCF, etc.

Step 2 — Cancel common factors

A factor in any numerator cancels with an identical factor in any denominator (across the entire product, not just within one fraction).

Step 3 — Multiply remaining factors

Multiply the remaining numerator factors together and the remaining denominator factors together. The result is the simplified product.

Example

Simplify \(\color{blue}{\frac{(x^{2} – 4)}{(x + 3)} \times \frac{(x + 3)}{(x – 2)}}\).

Factor: \(\color{blue}{\frac{((x + 2)(x – 2))}{(x + 3)} \times \frac{(x + 3)}{(x – 2)}}\)
Cancel (\(\color{blue}{x + 3}\)) and (\(\color{blue}{x – 2}\)):
= \(\color{blue}{(x + 2) \times \frac{1}{(1 \times 1)}}\) = \(\color{blue}{x + 2}\)  (x ≠ −3, x ≠ 2)

Step-by-Step Summary

1. Factor every numerator and denominator completely.
2. Write the full product as one fraction: all numerators on top, all denominators on bottom.
3. Cancel any factor that appears in both a numerator position and a denominator position.
4. Multiply the remaining factors.
5. State domain restrictions (all x-values that made any denominator zero).

Watch: Multiplying Rational Expressions (Video Lesson)

The Organic Chemistry Tutor explains the factor-and-cancel method for multiplying rational expressions with several worked examples:

Multiplying Rational Expressions – Worked Examples

Example 1: Simplify \(\color{blue}{\frac{(x^{2} – 4)}{(x + 3)} \times \frac{(x + 3)}{(x – 2)}}\).

Factor: \(\color{blue}{\frac{((x + 2)(x – 2))}{(x + 3)} \times \frac{(x + 3)}{(x – 2)}}\)
Cancel (\(\color{blue}{x + 3}\)) and (\(\color{blue}{x – 2}\)): \(\color{blue}{x + 2}\)  (x ≠ −3, x ≠ 2)

Example 2: Simplify \(\color{blue}{\frac{3x}{(x^{2} – 1)} \times \frac{(x + 1)}{6}}\).

Factor: \(\color{blue}{\frac{3x}{((x + 1)(x – 1))} \times \frac{(x + 1)}{6}}\)
Cancel (\(\color{blue}{x + 1}\)) and 3: = \(\color{blue}{\frac{x}{(2(x – 1))}}\) = \(\color{blue}{\frac{x}{(2x – 2)}}\)  (x ≠ 1, x ≠ −1)

Example 3: Simplify \(\color{blue}{\frac{(x^{2} – 9)}{(x + 1)} \times \frac{(x + 1)}{(x + 3)}}\).

Factor: \(\color{blue}{\frac{((x + 3)(x – 3))}{(x + 1)} \times \frac{(x + 1)}{(x + 3)}}\)
Cancel (\(\color{blue}{x + 3}\)) and (\(\color{blue}{x + 1}\)): \(\color{blue}{x – 3}\)  (x ≠ −1, x ≠ −3)

Example 4: Simplify \(\color{blue}{\frac{2x}{(x^{2} – 4)} \times \frac{(x – 2)}{4}}\).

Factor: \(\color{blue}{\frac{2x}{((x + 2)(x – 2))} \times \frac{(x – 2)}{4}}\)
Cancel (\(\color{blue}{x – 2}\)) and 2: = \(\color{blue}{\frac{x}{(2(x + 2))}}\) = \(\color{blue}{\frac{x}{(2x + 4)}}\)  (x ≠ 2, x ≠ −2)

More Practice: Multiplying Rational Expressions (Khan Academy)

Khan Academy explains multiplying rational expressions at the precalculus level with clear notation and practice:

Exercises for Multiplying Rational Expressions

1. \(\color{blue}{\frac{(x + 2)}{(x – 1)} \times \frac{(x – 1)}{(x + 4)}}\)
2. \(\color{blue}{\frac{(x^{2} – 1)}{(x + 2)} \times \frac{(x + 2)}{(x + 1)}}\)
3. \(\color{blue}{\frac{(4x)}{(x + 3)} \times \frac{(x + 3)}{(2)}}\)
4. \(\color{blue}{\frac{(x^{2} – 9)}{(x + 4)} \times \frac{(x + 4)}{(x – 3)}}\)
5. \(\color{blue}{\frac{(3x^{2})}{(x – 2)} \times \frac{(x – 2)}{(6x)}}\)

Answers

1. Cancel (\(\color{blue}{x – 1}\)): \(\color{blue}{\frac{(x + 2)}{(x + 4)}}\)
2. x² − \(\color{blue}{1 = (x + 1)(x – 1)}\); cancel (\(\color{blue}{x + 2}\)) and (\(\color{blue}{x + 1}\)): \(\color{blue}{x – 1}\)
3. Cancel (\(\color{blue}{x + 3}\)): \(\color{blue}{\frac{4x}{2}}\) = 2x
4. \(\color{blue}{(x + 3)(x – 3) = x}\)\(\color{blue}{^{2} – 9}\); cancel (\(\color{blue}{x + 4}\)) and (\(\color{blue}{x – 3}\)): \(\color{blue}{x + 3}\)
5. Cancel (\(\color{blue}{x – 2}\)) and 3x: \(\color{blue}{\frac{3x^{2}}{(6x)}}\) = \(\color{blue}{\frac{x}{2}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Multiplying Rational Expressions 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 Multiplying Polynomials Worksheet
• Download Factoring Trinomials x2 bx c Worksheet

Frequently Asked Questions

What is the rule for multiplying rational expressions?

Multiply the numerators together and the denominators together, just as with number fractions: (a/b) × (c/d) = ac/bd. The smart way to apply it is to factor everything first and cancel common factors before multiplying, so the numbers stay small. Remember that any value that makes an original denominator zero is excluded from the answer.

What is the first step in multiplying rational expressions?

The first step is to factor every numerator and denominator completely. Factoring shows you which pieces can cancel. For example, x² − 4 becomes (x − 2)(x + 2) and 3x + 6 becomes 3(x + 2). Once you see the shared (x + 2), you can divide it out. Write down restrictions now too, while the original denominators are in front of you.

How do you multiply rational expressions step by step?

Factor, cancel, multiply, then state restrictions. For (x² − 4)/(3x) × 6x²/(x + 2), factor x² − 4 as (x − 2)(x + 2). Cancel (x + 2) from the top and bottom, then reduce 6x²/(3x) to 2x. What remains is 2x(x − 2), or 2x² − 4x. Because 3x and x + 2 were denominators, x ≠ 0 and x ≠ −2.

Can you cross multiply rational expressions?

Not when you are multiplying them. Cross multiplication is for solving an equation like a/b = c/d, where you write ad = bc. When you multiply two rational expressions, you multiply straight across instead. What you can do is cross-cancel: a factor in the first numerator may cancel with the same factor in the second denominator, such as the x in x/5 × 10/x², which leaves 2/x.

How do you multiply rational expressions with multiple variables?

Handle the numbers and each variable separately, using exponent rules to cancel. For (4a²b)/(9c) × (3c²)/(2ab³), multiply to get 12a²bc²/(18ab³c). Reduce 12/18 to 2/3, then a²/a = a, b/b³ = 1/b², and c²/c = c. The answer is 2ac/(3b²), with a, b and c not equal to 0.

Related Topics

• Dividing Rational Expressions
• Adding and Subtracting Rational Expressions
• Simplifying Complex Fractions
• Graphing Rational Functions
• Multiplying and Dividing Functions