# How to Multiply Rational Expressions

A rational expression is a ratio of two polynomials and there is a simple way to multiply these expressions, which we will teach you in this article with examples.

## Method of Multiplying Rational Expressions

• Multiplying rational expressions is the same as multiplying fractions. First, multiply numerators and then multiply denominators. Then, simplify as needed.

## Examples

### Multiplying Rational Expressions – Example 1:

Solve: $$\frac{x + 6}{x- 1}×\frac{x – 1}{5}=$$

Solution:

Multiply numerators and denominators: $$\frac{a}{b}×\frac{c}{d}=\frac{a×c}{b×d}$$
$$\frac{x+6}{x-1}×\frac{x-1}{5}=\frac{(x+6)(x-1)}{5(x- 1)}$$
Cancel the common factor: $$(x- 1)$$
Then: $$\frac{(x + 6)(x-1)}{5(x- 1)}=\frac{(x + 6)}{5}$$

### Multiplying Rational Expressions – Example 2:

Solve: $$\frac{x – 2}{x+3}×\frac{2x + 6}{x – 2}=$$

Solution:

Multiply numerators and denominators: $$\frac{x – 2}{x+3}×\frac{2x + 6}{x – 2}=\frac{(x – 2)(2x + 6)}{(x+3)(x-2)}$$
Cancel the common factor: $$\frac{(x – 2)(2x + 6)}{(x+3)(x-2)}=\frac{(2x + 6)}{(x+3)}$$
Factor $$2x+6=2(x+3)$$
Then: $$\frac{2(x+3)}{(x+3)}=2$$

### Multiplying Rational Expressions – Example 3:

Solve: $$\frac{x + 5}{x- 1}×\frac{x – 1}{3}=$$

Solution:

Multiply fractions: $$\frac{x + 5}{x- 1}×\frac{x – 1}{3}=\frac{(x + 5)(x-1)}{3(x- 1)}$$
Cancel the common factor: $$(x- 1)$$,then: $$\frac{(x + 5)(x-1)}{3(x- 1)}=\frac{(x + 5)}{3}$$

### Multiplying Rational Expressions – Example 4:

Solve: $$\frac{x – 5}{x+4}×\frac{2x + 8}{x – 5}=$$

Solution:

Multiply fractions: $$\frac{x – 5}{x+4}×\frac{2x + 8}{x – 5}=\frac{(x – 5)(2x + 8)}{(x+4)(x-5)}$$
Cancel the common factor: $$\frac{(x – 5)(2x + 8)}{(x+4)(x-5)}=\frac{(2x + 8)}{(x+4)}$$
Factor $$2x+8=2(x+4)$$, Then: $$\frac{2(x+4)}{(x+4)}=2$$

## Exercises for Multiplying Rational Expressions

### Multiply Rational Expressions.

1. $$\color{blue}{\frac{x-4}{x+3}×\frac{x + 6}{x – 4}=}$$
2. $$\color{blue}{\frac{x+5}{x+2}×\frac{x }{x+5}=}$$
3. $$\color{blue}{\frac{x+9}{x}×\frac{x^2 }{x+9}=}$$
4. $$\color{blue}{\frac{3x^2}{10}×\frac{5 }{4x}=}$$
5. $$\color{blue}{\frac{x+2}{5x}×\frac{6 }{6x+12}=}$$
6. $$\color{blue}{\frac{x-3}{3x+4}×\frac{9x+12 }{x-3}=}$$
1. $$\color{blue}{\frac{x+6}{x+3}}$$
2. $$\color{blue}{\frac{x}{x+2}}$$
3. $$\color{blue}{x}$$
4. $$\color{blue}{\frac{3x}{8}}$$
5. $$\color{blue}{\frac{1}{5x}}$$
6. $$\color{blue}{3}$$

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