Multiplying Algebraic Fractions

The product of two algebraic fractions is itself an algebraic fraction. Its numerator is the product of the numerators, and its denominator is the product of the denominators: $$ \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} $$

How to multiply algebraic fractions

When multiplying algebraic fractions, the procedure is straightforward:

Factorize both numerators and denominators.
Simplify wherever possible - either within the same fraction (vertical simplification) or across fractions (cross simplification).
Multiply the numerators together and the denominators together.

The result is the algebraic fraction representing the product.

Note. After finding the product, you may still need to reduce the fraction to its simplest form. It’s easy to overlook a simplification step earlier in the process.

Worked examples

Example 1

Let’s start with this product:

$$ \frac{2x}{3y} \cdot \frac{4y}{3z} $$

First, simplify vertically and across the fractions:

$$ \frac{2x}{3 \require{cancel} \cancel{y}} \cdot \frac{4 \cancel{y}}{3z} $$

$$ \frac{2x}{3} \cdot \frac{4}{3z} $$

Now multiply numerators and denominators:

$$ \frac{2x \cdot 4}{3 \cdot 3z} $$

$$ \frac{8x}{9z} $$

The final result is the algebraic fraction representing the product.

Example 2

Now consider the following product:

$$ \frac{4x^2}{x^2-y^2} \cdot \frac{x+y}{2x} $$

First, factorize wherever possible:

$$ \frac{2^2x^2}{x^2-y^2} \cdot \frac{x+y}{2x} $$

The denominator of the first fraction is a difference of squares:

$$ \frac{2^2x^2}{(x-y)(x+y)} \cdot \frac{x+y}{2x} $$

Next, simplify vertically and across the fractions:

$$ \frac{2^2x^2}{(x-y)\require{cancel}\cancel{(x+y)}} \cdot \frac{\cancel{x+y}}{2x} $$

$$ \frac{2^{\cancel{2}}x^2}{x-y} \cdot \frac{1}{\cancel{2}x} $$

$$ \frac{2x^\cancel{2}}{x-y} \cdot \frac{1}{\cancel{x}} $$

$$ \frac{2x}{x-y} \cdot \frac{1}{1} $$

Finally, multiply numerators and denominators:

$$ \frac{2x \cdot 1}{(x-y) \cdot 1} $$

$$ \frac{2x}{x-y} $$

The result is the product in irreducible form.

And the process continues in the same way.