Multiplying and Dividing Rational Expressions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Multiplying and Dividing Rational Expressions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Multiplying rational expressions by multiplying numerators together and denominators together (after factoring and canceling). Dividing by multiplying by the reciprocal of the divisor.

It works exactly like multiplying and dividing numeric fractions. To multiply: factor everything, cancel common factors across any numerator and any denominator, then multiply across. To divide: flip the second fraction and multiply. \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factor first, cancel common factors across all numerators and denominators, then multiply what remains.

Common stuck point: When dividing, remember to flip the SECOND fraction (the divisor), not the first. Then proceed as with multiplication.

Sense of Study hint: Factor everything first, cancel common factors across all numerators and denominators, then multiply what remains.

Worked Examples

Example 1

medium
Multiply \frac{x^2 - 1}{x + 3} \cdot \frac{x + 3}{x + 1}.

Solution

  1. 1
    Step 1: Factor: \frac{(x+1)(x-1)}{x+3} \cdot \frac{x+3}{x+1}.
  2. 2
    Step 2: Cancel (x+3) and (x+1): x - 1.
  3. 3
    Step 3: Restrictions: x \neq -3, -1.
  4. 4
    Check: At x = 2: \frac{3}{5} \cdot \frac{5}{3} = 1 and 2-1 = 1 ✓

Answer

x - 1, x \neq -3, -1
When multiplying rational expressions, factor everything first, cancel common factors across numerators and denominators, then multiply what remains.

Example 2

hard
Divide \frac{x^2 - 4}{x^2 + x} \div \frac{x - 2}{x}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Multiply \frac{3}{x} \cdot \frac{x^2}{6}.

Example 2

medium
Divide \frac{x + 5}{x - 1} \div \frac{x + 5}{x^2 - 1}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

simplifying rational expressionsfactoring