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.

## Related Topics

- How to Add and Subtract Rational Expressions
- How to Divide Rational Expressions
- How to Solve Rational Equations
- How to Simplify Complex Fractions
- How to Graph Rational Expressions

## 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.

- \(\color{blue}{\frac{x-4}{x+3}×\frac{x + 6}{x – 4}=}\)
- \(\color{blue}{\frac{x+5}{x+2}×\frac{x }{x+5}=}\)
- \(\color{blue}{\frac{x+9}{x}×\frac{x^2 }{x+9}=}\)
- \(\color{blue}{\frac{3x^2}{10}×\frac{5 }{4x}=}\)
- \(\color{blue}{\frac{x+2}{5x}×\frac{6 }{6x+12}=}\)
- \(\color{blue}{\frac{x-3}{3x+4}×\frac{9x+12 }{x-3}=}\)

- \(\color{blue}{\frac{x+6}{x+3}}\)
- \(\color{blue}{\frac{x}{x+2}}\)
- \(\color{blue}{x}\)
- \(\color{blue}{\frac{3x}{8}}\)
- \(\color{blue}{\frac{1}{5x}}\)
- \(\color{blue}{3}\)