Solved Rational Expression Worksheet #5: Multiplying & | Chegg.com - Free Printable

Let's solve each problem step-by-step, factoring where necessary and showing all work. We'll be multiplying or dividing rational expressions.

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1.

$$
\frac{2a^2b}{b^2c} \div \frac{b}{a}
$$

Step 1: Rewrite division as multiplication by the reciprocal:
$$
\frac{2a^2b}{b^2c} \cdot \frac{a}{b}
$$

Step 2: Simplify by canceling common factors:

- $ a^2 \cdot a = a^3 $
- $ b \cdot b = b^2 $ in denominator (from $ b^2 $ and extra $ b $)
- So: $ \frac{2a^3}{b^2 c} $

But let’s write it clearly:

Numerator: $ 2a^2b \cdot a = 2a^3b $
Denominator: $ b^2c \cdot b = b^3c $

So:
$$
\frac{2a^3b}{b^3c} = \frac{2a^3}{b^2c}
$$

✔ Answer: $ \boxed{\frac{2a^3}{b^2c}} $

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2.

$$
\frac{y^2 - 2y - 15}{4} \cdot \frac{8}{y + 3}
$$

Step 1: Factor numerator of first expression:
$$
y^2 - 2y - 15 = (y - 5)(y + 3)
$$

So:
$$
\frac{(y - 5)(y + 3)}{4} \cdot \frac{8}{y + 3}
$$

Step 2: Cancel $ y + 3 $ (assuming $ y \neq -3 $):
$$
\frac{(y - 5)}{4} \cdot \frac{8}{1} = \frac{8(y - 5)}{4} = 2(y - 5)
$$

✔ Answer: $ \boxed{2(y - 5)} $

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3.

$$
\frac{x - 5}{6} \div \frac{2x - 10}{12}
$$

Step 1: Rewrite as multiplication:
$$
\frac{x - 5}{6} \cdot \frac{12}{2x - 10}
$$

Step 2: Factor $ 2x - 10 = 2(x - 5) $

So:
$$
\frac{x - 5}{6} \cdot \frac{12}{2(x - 5)}
$$

Cancel $ x - 5 $, and simplify numbers:
- $ \frac{12}{6 \cdot 2} = \frac{12}{12} = 1 $

So:
$$
1
$$

✔ Answer: $ \boxed{1} $

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4.

$$
\frac{5n + 15}{4n + 8} \cdot \frac{2n + 4}{3n + 9}
$$

Step 1: Factor all terms:

- $ 5n + 15 = 5(n + 3) $
- $ 4n + 8 = 4(n + 2) $
- $ 2n + 4 = 2(n + 2) $
- $ 3n + 9 = 3(n + 3) $

Now:
$$
\frac{5(n + 3)}{4(n + 2)} \cdot \frac{2(n + 2)}{3(n + 3)}
$$

Step 2: Cancel common factors:

- $ n + 3 $ cancels
- $ n + 2 $ cancels
- $ 5 \cdot 2 = 10 $, $ 4 \cdot 3 = 12 $

So:
$$
\frac{10}{12} = \frac{5}{6}
$$

✔ Answer: $ \boxed{\frac{5}{6}} $

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5.

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

Step 1: Factor:

- $ x^2 - 2x = x(x - 2) $
- $ 3x - 6 = 3(x - 2) $

So:
$$
\frac{x(x - 2)}{6} \div \frac{3(x - 2)}{x}
= \frac{x(x - 2)}{6} \cdot \frac{x}{3(x - 2)}
$$

Step 2: Cancel $ x - 2 $, and simplify:

- $ x \cdot x = x^2 $
- Denominator: $ 6 \cdot 3 = 18 $

So:
$$
\frac{x^2}{18}
$$

✔ Answer: $ \boxed{\frac{x^2}{18}} $

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6.

$$
\frac{m^2 - 2m - 8}{8m + 24} \div \frac{2m - 8}{m^2 + 7m + 12}
$$

Step 1: Factor all expressions:

- $ m^2 - 2m - 8 = (m - 4)(m + 2) $
- $ 8m + 24 = 8(m + 3) $
- $ 2m - 8 = 2(m - 4) $
- $ m^2 + 7m + 12 = (m + 3)(m + 4) $

So:
$$
\frac{(m - 4)(m + 2)}{8(m + 3)} \div \frac{2(m - 4)}{(m + 3)(m + 4)}
= \frac{(m - 4)(m + 2)}{8(m + 3)} \cdot \frac{(m + 3)(m + 4)}{2(m - 4)}
$$

Step 2: Cancel:

- $ m - 4 $ cancels
- $ m + 3 $ cancels
- $ 8 \cdot 2 = 16 $

Left with:
$$
\frac{(m + 2)(m + 4)}{16}
$$

✔ Answer: $ \boxed{\frac{(m + 2)(m + 4)}{16}} $

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7.

$$
\frac{x + 3}{10x + 20} \div \frac{x + 2}{x^2 + 4x + 3}
$$

Step 1: Factor:

- $ 10x + 20 = 10(x + 2) $
- $ x^2 + 4x + 3 = (x + 1)(x + 3) $

So:
$$
\frac{x + 3}{10(x + 2)} \div \frac{x + 2}{(x + 1)(x + 3)}
= \frac{x + 3}{10(x + 2)} \cdot \frac{(x + 1)(x + 3)}{x + 2}
$$

Step 2: Multiply:

Numerator: $ (x + 3)(x + 1)(x + 3) = (x + 3)^2(x + 1) $
Denominator: $ 10(x + 2)(x + 2) = 10(x + 2)^2 $

So:
$$
\frac{(x + 3)^2(x + 1)}{10(x + 2)^2}
$$

✔ Answer: $ \boxed{\frac{(x + 3)^2(x + 1)}{10(x + 2)^2}} $

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8.

$$
\frac{x^2 - x - 12}{x - 4} \div \frac{2x + 6}{x - 5}
$$

Step 1: Factor:

- $ x^2 - x - 12 = (x - 4)(x + 3) $
- $ 2x + 6 = 2(x + 3) $

So:
$$
\frac{(x - 4)(x + 3)}{x - 4} \div \frac{2(x + 3)}{x - 5}
= (x + 3) \div \frac{2(x + 3)}{x - 5}
$$

(Since $ x - 4 $ cancels)

Now:
$$
(x + 3) \cdot \frac{x - 5}{2(x + 3)} = \frac{x - 5}{2}
$$

✔ Answer: $ \boxed{\frac{x - 5}{2}} $

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9.

$$
\frac{x^2 - 5x - 6}{5x + 15} \div \frac{x^2 - 3x - 4}{7x + 21}
$$

Step 1: Factor:

- $ x^2 - 5x - 6 = (x - 6)(x + 1) $
- $ 5x + 15 = 5(x + 3) $
- $ x^2 - 3x - 4 = (x - 4)(x + 1) $
- $ 7x + 21 = 7(x + 3) $

So:
$$
\frac{(x - 6)(x + 1)}{5(x + 3)} \div \frac{(x - 4)(x + 1)}{7(x + 3)}
= \frac{(x - 6)(x + 1)}{5(x + 3)} \cdot \frac{7(x + 3)}{(x - 4)(x + 1)}
$$

Step 2: Cancel:

- $ x + 1 $ cancels
- $ x + 3 $ cancels
- Left: $ \frac{(x - 6)}{5} \cdot \frac{7}{(x - 4)} = \frac{7(x - 6)}{5(x - 4)} $

✔ Answer: $ \boxed{\frac{7(x - 6)}{5(x - 4)}} $

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10.

$$
\frac{24x^3}{25y^3} \cdot \frac{15y^2}{8x^2}
$$

Step 1: Multiply numerators and denominators:

Numerator: $ 24x^3 \cdot 15y^2 = 360x^3y^2 $
Denominator: $ 25y^3 \cdot 8x^2 = 200x^2y^3 $

So:
$$
\frac{360x^3y^2}{200x^2y^3}
$$

Simplify:

- $ \frac{360}{200} = \frac{9}{5} $
- $ x^3 / x^2 = x $
- $ y^2 / y^3 = 1/y $

So:
$$
\frac{9x}{5y}
$$

✔ Answer: $ \boxed{\frac{9x}{5y}} $

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11.

$$
\frac{6x - 18}{4x} \cdot \frac{x}{2x - 6}
$$

Step 1: Factor:

- $ 6x - 18 = 6(x - 3) $
- $ 2x - 6 = 2(x - 3) $

So:
$$
\frac{6(x - 3)}{4x} \cdot \frac{x}{2(x - 3)}
$$

Step 2: Cancel:

- $ x $ cancels
- $ x - 3 $ cancels
- $ 6 / (4 \cdot 2) = 6 / 8 = 3/4 $

So:
$$
\frac{3}{4}
$$

✔ Answer: $ \boxed{\frac{3}{4}} $

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12.

$$
\frac{3x + 12}{12x} \div \frac{x + 4}{48x^3}
$$

Step 1: Factor:

- $ 3x + 12 = 3(x + 4) $
- $ 12x = 12x $
- $ x + 4 $ stays
- $ 48x^3 = 48x^3 $

So:
$$
\frac{3(x + 4)}{12x} \div \frac{x + 4}{48x^3}
= \frac{3(x + 4)}{12x} \cdot \frac{48x^3}{x + 4}
$$

Step 2: Cancel:

- $ x + 4 $ cancels
- $ 3 \cdot 48 = 144 $
- $ 12 \cdot 1 = 12 $
- $ x^3 / x = x^2 $

So:
$$
\frac{144x^2}{12} = 12x^2
$$

✔ Answer: $ \boxed{12x^2} $

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✔ Final Answers Summary:

1. $ \frac{2a^3}{b^2c} $
2. $ 2(y - 5) $
3. $ 1 $
4. $ \frac{5}{6} $
5. $ \frac{x^2}{18} $
6. $ \frac{(m + 2)(m + 4)}{16} $
7. $ \frac{(x + 3)^2(x + 1)}{10(x + 2)^2} $
8. $ \frac{x - 5}{2} $
9. $ \frac{7(x - 6)}{5(x - 4)} $
10. $ \frac{9x}{5y} $
11. $ \frac{3}{4} $
12. $ 12x^2 $

Let me know if you'd like these formatted in a worksheet-style table!