Multiplying and Dividing Rational Expressions Worksheets - Math Monks - Free Printable

1. $(2x + 4) \cdot \frac{3x}{3x + 6}$

Factor numerator and denominator:
$(2x + 4) = 2(x + 2)$
$3x + 6 = 3(x + 2)$

So the expression becomes:
$2(x + 2) \cdot \frac{3x}{3(x + 2)}$

Cancel common factors:
- $x + 2$ cancels
- $3$ cancels

Left with: $2 \cdot x = 2x$

Final answer: $2x$

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2. $(3x + 6) \cdot \frac{2x}{2x + 4}$

Factor:
$3x + 6 = 3(x + 2)$
$2x + 4 = 2(x + 2)$

Expression becomes:
$3(x + 2) \cdot \frac{2x}{2(x + 2)}$

Cancel common factors:
- $x + 2$ cancels
- $2$ cancels

Left with: $3 \cdot x = 3x$

Final answer: $3x$

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3. $\frac{x^2 - 4}{x^2 - 1} \cdot \frac{x + 1}{x + 2}$

Factor:
- $x^2 - 4 = (x - 2)(x + 2)$
- $x^2 - 1 = (x - 1)(x + 1)$

Expression becomes:
$\frac{(x - 2)(x + 2)}{(x - 1)(x + 1)} \cdot \frac{x + 1}{x + 2}$

Cancel common factors:
- $x + 2$ cancels
- $x + 1$ cancels

Left with: $\frac{x - 2}{x - 1}$

Final answer: $\frac{x - 2}{x - 1}$

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4. $\frac{x^2 - 1}{x - 3} \cdot \frac{x^2 - 9}{x + 1}$

Factor:
- $x^2 - 1 = (x - 1)(x + 1)$
- $x^2 - 9 = (x - 3)(x + 3)$

Expression becomes:
$\frac{(x - 1)(x + 1)}{x - 3} \cdot \frac{(x - 3)(x + 3)}{x + 1}$

Cancel common factors:
- $x - 3$ cancels
- $x + 1$ cancels

Left with: $(x - 1)(x + 3)$

Final answer: $(x - 1)(x + 3)$

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5. $\frac{x^2 - x - 12}{3x - 9} \div \frac{x - 4}{12}$

First, rewrite division as multiplication by reciprocal:
$\frac{x^2 - x - 12}{3x - 9} \cdot \frac{12}{x - 4}$

Factor:
- $x^2 - x - 12 = (x - 4)(x + 3)$
- $3x - 9 = 3(x - 3)$

Expression becomes:
$\frac{(x - 4)(x + 3)}{3(x - 3)} \cdot \frac{12}{x - 4}$

Cancel common factors:
- $x - 4$ cancels
- $12$ and $3$: $12/3 = 4$

Left with: $4 \cdot \frac{x + 3}{x - 3} = \frac{4(x + 3)}{x - 3}$

Final answer: $\frac{4(x + 3)}{x - 3}$

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6. $\frac{x^2}{6 - 3x} \div \frac{x}{8 - 4x}$

Rewrite division as multiplication by reciprocal:
$\frac{x^2}{6 - 3x} \cdot \frac{8 - 4x}{x}$

Factor:
- $6 - 3x = -3(x - 2)$
- $8 - 4x = -4(x - 2)$

Expression becomes:
$\frac{x^2}{-3(x - 2)} \cdot \frac{-4(x - 2)}{x}$

Cancel common factors:
- $x$ cancels (one $x$ from numerator and denominator)
- $x - 2$ cancels
- $-3$ and $-4$: $(-4)/(-3) = 4/3$

Left with: $\frac{x \cdot 4}{3} = \frac{4x}{3}$

Final answer: $\frac{4x}{3}$

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7. $\frac{x - 4}{x^2 - 5x + 6} \div \frac{x - 3}{x^2 - 6x + 9}$

Rewrite division as multiplication by reciprocal:
$\frac{x - 4}{x^2 - 5x + 6} \cdot \frac{x^2 - 6x + 9}{x - 3}$

Factor:
- $x^2 - 5x + 6 = (x - 2)(x - 3)$
- $x^2 - 6x + 9 = (x - 3)^2$

Expression becomes:
$\frac{x - 4}{(x - 2)(x - 3)} \cdot \frac{(x - 3)^2}{x - 3}$

Simplify second fraction: $\frac{(x - 3)^2}{x - 3} = x - 3$

Now: $\frac{x - 4}{(x - 2)(x - 3)} \cdot (x - 3)$

Cancel $x - 3$:
$\frac{x - 4}{x - 2}$

Final answer: $\frac{x - 4}{x - 2}$

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8. $\frac{x^2 - 9}{6x} \div \frac{x^2 - 9}{18x^3}$

Rewrite division as multiplication by reciprocal:
$\frac{x^2 - 9}{6x} \cdot \frac{18x^3}{x^2 - 9}$

Cancel common factors:
- $x^2 - 9$ cancels
- $6x$ and $18x^3$: $18x^3 / 6x = 3x^2$

Left with: $3x^2$

Final answer: $3x^2$