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

Do I need a common denominator to multiply rational expressions?

No. A common denominator is only needed for addition and subtraction. To multiply, you simply multiply numerators together and denominators together (after factoring and cancelling).

Can I cancel factors diagonally (across two different fractions)?

Yes. When multiplying two fractions, a factor in either numerator can cancel with the same factor in either denominator — whether they are in the same fraction or not. This is sometimes called “cross-cancelling.”

Why must I state domain restrictions in the final answer?

Cancelling a factor hides the original restriction from the simplified expression. For example, if you cancel (\(\color{blue}{x + 3}\)), the simplified answer no longer shows x = −3 as a problem — but it still is, because the original denominator was undefined there.

Related Topics

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