To multiply two or more algebraic fractions, we simply multiply the numerators with the numerators and the denominators with the denominators. Then, we simplify the final fraction by reducing common factors.

Here, we will learn the steps that we can apply to solve a multiplication of algebraic fractions. Then, we will look at some practice examples.

## Steps for multiplying two or more algebraic fractions

Two or more algebraic fractions can be multiplied by multiplying their numerators and denominators separately.

For example, if we have the fractions $latex \frac{a}{b}$ and $latex \frac{x}{y}$, we can multiply them in the following way:

$$\frac{a}{b}\times \frac{x}{y}=\frac{a\times x}{b\times y}$$

Then, we can find the product of two or more algebraic fractions by following the steps below:

#### 1. Multiply the numerators.

#### 2. Multiply the denominators

#### 3. Simplify the resulting fraction.

Reduce the common factors in the numerator and denominator.

## Solved examples of multiplication of algebraic fractions

**EXAMPLE **1

**EXAMPLE**

Find the product of $latex \frac{2x}{y}$ and $latex \frac{x}{y}$.

##### Solution

To multiply these fractions, we multiply the numerators and denominators separately. Then, we have:

$$\frac{2x}{y} \times \frac{x}{y}=\frac{(2x)(x)}{(y)(y)}$$

$$=\frac{2x^2}{y^2}$$

We have no common factors between the numerator and denominator, so we can no longer simplify.

**EXAMPLE **2

**EXAMPLE**Multiply the fractions $latex \frac{x+1}{5}$ and $latex \frac{x}{10}$.

##### Solution

Multiplying the numerators and denominators separately, we have:

$$\frac{x+1}{5} \times \frac{x}{10}=\frac{(x+1)(x)}{(5)(10)}$$

$$=\frac{x^2+x}{50}$$

This is the resulting fraction. Alternatively, we can write it by taking out a common factor in the numerator:

$$=\frac{x(x+1)}{50}$$

**EXAMPLE **3

**EXAMPLE**Multiply the fractions $latex \frac{x+5}{2x+2}$ and $latex \frac{2x}{y}$.

##### Solution

Multiplying the numerators and denominators separately, we have:

$$\frac{x+5}{2x+2} \times \frac{2x}{y}=\frac{(x+5)(2x)}{(2x+2)(y)}$$

$$=\frac{2x^2+10x}{2xy+2y}$$

We can simplify the fraction by considering common factors:

$$=\frac{2x(x+5)}{2y(x+1)}$$

$$=\frac{x(x+5)}{y(x+1)}$$

**EXAMPLE **4

**EXAMPLE**Find the product of the following multiplication:

$$ \frac{x^2-2x}{2x-10} \cdot \frac{2x+4}{x+5} $$

##### Solution

Multiplying the numerators and denominators, we have:

$$ \frac{(x^2-2x)(2x+4)}{(2x-10)(x+5)} $$

When we multiply this, we have:

$$ =\frac{2x^3+4x^2-4x^2-8x}{2x^2+10x-10x-50} $$

Combining like terms, we have:

$$ =\frac{2x^3-8x}{2x^2-50} $$

We can simplify by taking common factors:

$$ =\frac{2x(x^2-4)}{2(x^2-25)} $$

$$ =\frac{x(x^2-4)}{x^2-25} $$

**EXAMPLE **5

**EXAMPLE**What is the result of the following multiplication of fractions?

$$ \frac{a^2-4a – 5}{2a^2-50} \cdot \frac{2a-2}{3a+3} $$

##### Solution

When we multiply the numerators and the denominators, we have:

$$ \frac{(a^2-4a – 5)(2a-2)}{(2a^2-50)(3a+3)} $$

We can factor the numerator of the first fraction as follows:

$$ (a^2-4a – 5) = (a-5)(a+1) $$

and we take out the common factor wherever possible:

$$ =\frac{(a-5)(a+1)2(a-1)}{(2(a^2-25))3(a+1)} $$

Using the difference of squares, we can write $latex (a^2-25)$ as $latex (a – 5)(a+5) $.

$$ =\frac{(a-5)(a+1)2(a-1)}{(2(a – 5)(a+5))3(a+1)} $$

The fraction can be simplified to get the following:

$$ =\frac{(a-1)}{(a+5)3} $$

The end result is:

$$ =\frac{(a-1)}{3(a+5)} $$

**EXAMPLE **6

**EXAMPLE**Find the product of the following algebraic fractions:

$$ \frac{5}{x} \cdot \frac{2x}{b^2} \cdot \frac{3b}{10} $$

##### Solution

In this case, we have a multiplication of three algebraic fractions. However, the process to follow is the same.

We just multiply the numerators and denominators separately:

$$ =\frac{(5)(2x)(3b)}{(x)(b^2)(10)} $$

$$ =\frac{30xb}{10xb^2} $$

After simplifying, we get the following:

$$ =\frac{3}{b} $$

You can explore additional examples on this topic in our article: Multiplying Algebraic Fractions – Examples and Practice.

## Multiplication of Algebraic Fractions – Practice problems

#### What is the result of multiplying the fractions and simplifying? $$\frac{5x^2}{x^2-2x} \cdot \frac{x^2-4}{x^2+2x} $$

Write the answer in the input box.

## See also

Interested in learning more about algebraic fractions? You can take a look at these pages:

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