Worksheet #8 for simplifying, multiplying, and dividing rational expressions.

Let's solve each problem step-by-step from the Rational Expression Worksheet #8: Simplify/Multiply/Divide. We'll simplify, multiply, or divide rational expressions by factoring when necessary and canceling common factors.

---

Simplify (remember to factor when necessary)

#### 1. $\frac{18x^6}{27x^4}$

- Simplify coefficients: $\frac{18}{27} = \frac{2}{3}$
- Simplify variables: $x^6 / x^4 = x^{6-4} = x^2$

✔ Answer: $\boxed{\frac{2x^2}{3}}$

---

#### 2. $\frac{x^2 + 6x + 8}{3x + 12}$

- Factor numerator: $x^2 + 6x + 8 = (x+2)(x+4)$
- Factor denominator: $3x + 12 = 3(x + 4)$

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

Cancel $(x+4)$ (as long as $x \ne -4$):

✔ Answer: $\boxed{\frac{x+2}{3}}$

---

#### 3. $\frac{x^2 - 7x + 12}{x^2 + 2x - 15}$

- Factor numerator: $x^2 - 7x + 12 = (x-3)(x-4)$
- Factor denominator: $x^2 + 2x - 15 = (x+5)(x-3)$

Now:
$$
\frac{(x-3)(x-4)}{(x+5)(x-3)}
$$

Cancel $(x-3)$ ($x \ne 3$):

✔ Answer: $\boxed{\frac{x-4}{x+5}}$

---

Multiply or Divide (remember to factor when necessary)

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

Factor all parts:

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

Now:
$$
\frac{x+3}{(x-2)^2} \cdot \frac{(x-2)(x+1)}{(x+1)(x+3)}
$$

Cancel common terms:
- $(x+3)$ cancels
- $(x+1)$ cancels
- One $(x-2)$ cancels

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

✔ Answer: $\boxed{\frac{1}{x-2}}$, $x \ne 2, -3, -1$

---

#### 5. $\frac{x^2 - x - 12}{3x + 9} \div \frac{x^2 + x - 20}{x + 5}$

First, rewrite division as multiplication by reciprocal:

$$
\frac{x^2 - x - 12}{3x + 9} \cdot \frac{x + 5}{x^2 + x - 20}
$$

Factor:

- $x^2 - x - 12 = (x-4)(x+3)$
- $3x + 9 = 3(x+3)$
- $x^2 + x - 20 = (x+5)(x-4)$

Now:
$$
\frac{(x-4)(x+3)}{3(x+3)} \cdot \frac{x+5}{(x+5)(x-4)}
$$

Cancel:
- $(x+3)$
- $(x+5)$
- $(x-4)$

Left with:
$$
\frac{1}{3}
$$

✔ Answer: $\boxed{\frac{1}{3}}$, $x \ne -3, -5, 4$

---

#### 6. $\frac{15x^2}{45x^3} \div \frac{5x^6}{9x^4}$

Rewrite as multiplication:
$$
\frac{15x^2}{45x^3} \cdot \frac{9x^4}{5x^6}
$$

Simplify each fraction:

- $\frac{15}{45} = \frac{1}{3}$, so $\frac{15x^2}{45x^3} = \frac{1}{3x}$
- $\frac{9x^4}{5x^6} = \frac{9}{5x^2}$

Now:
$$
\frac{1}{3x} \cdot \frac{9}{5x^2} = \frac{9}{15x^3} = \frac{3}{5x^3}
$$

✔ Answer: $\boxed{\frac{3}{5x^3}}$

---

#### 7. $\frac{6}{x^2 - 9x + 20} \cdot \frac{5x - 25}{3x - 6}$

Factor:

- $x^2 - 9x + 20 = (x-4)(x-5)$
- $5x - 25 = 5(x - 5)$
- $3x - 6 = 3(x - 2)$

Now:
$$
\frac{6}{(x-4)(x-5)} \cdot \frac{5(x-5)}{3(x-2)}
$$

Cancel:
- $(x-5)$
- $6$ and $3$: $6/3 = 2$

So:
$$
\frac{2 \cdot 5}{(x-4)(x-2)} = \frac{10}{(x-4)(x-2)}
$$

✔ Answer: $\boxed{\frac{10}{(x-4)(x-2)}}$, $x \ne 4, 5, 2$

---

#### 8. $\frac{6x - 12}{4x^2} \cdot \frac{3x^3}{2x - 4}$

Factor:

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

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

Cancel:
- $(x-2)$
- $6$ and $4$: $6/4 = 3/2$
- $3x^3$ and $x^2$: $x^3/x^2 = x$

So:
$$
\frac{3}{2} \cdot \frac{3x}{2} = \frac{9x}{4}
$$

Wait — let's do it carefully:

Numerator: $6(x-2) \cdot 3x^3 = 18x^3(x-2)$
Denominator: $4x^2 \cdot 2(x-2) = 8x^2(x-2)$

Now:
$$
\frac{18x^3(x-2)}{8x^2(x-2)} = \frac{18x^3}{8x^2} = \frac{18}{8}x = \frac{9}{4}x
$$

✔ Answer: $\boxed{\frac{9x}{4}}$, $x \ne 0, 2$

---

#### 9. $\frac{3x - 21}{x^2 - 3x - 28} \cdot \frac{5x + 20}{2x + 8}$

Factor:

- $3x - 21 = 3(x - 7)$
- $x^2 - 3x - 28 = (x - 7)(x + 4)$
- $5x + 20 = 5(x + 4)$
- $2x + 8 = 2(x + 4)$

Now:
$$
\frac{3(x-7)}{(x-7)(x+4)} \cdot \frac{5(x+4)}{2(x+4)}
$$

Cancel:
- $(x-7)$
- $(x+4)$ in first and second fraction
- Left with: $\frac{3}{x+4} \cdot \frac{5}{2(x+4)} = \frac{15}{2(x+4)^2}$

Wait — let’s recheck:

After cancellation:
- First fraction becomes: $\frac{3}{x+4}$
- Second: $\frac{5(x+4)}{2(x+4)} = \frac{5}{2}$

So total: $\frac{3}{x+4} \cdot \frac{5}{2} = \frac{15}{2(x+4)}$

But wait: denominator has two $(x+4)$s? Let's see:

Actually:
- First: $\frac{3(x-7)}{(x-7)(x+4)} = \frac{3}{x+4}$
- Second: $\frac{5(x+4)}{2(x+4)} = \frac{5}{2}$

So product: $\frac{3}{x+4} \cdot \frac{5}{2} = \frac{15}{2(x+4)}$

✔ Answer: $\boxed{\frac{15}{2(x+4)}}$, $x \ne 7, -4$

---

#### 10. $\frac{x^2 - 5x - 6}{2x + 6} \div \frac{x^2 - 3x - 4}{4x + 12}$

Rewrite as multiplication:

$$
\frac{x^2 - 5x - 6}{2x + 6} \cdot \frac{4x + 12}{x^2 - 3x - 4}
$$

Factor:

- $x^2 - 5x - 6 = (x-6)(x+1)$
- $2x + 6 = 2(x+3)$
- $4x + 12 = 4(x+3)$
- $x^2 - 3x - 4 = (x-4)(x+1)$

Now:
$$
\frac{(x-6)(x+1)}{2(x+3)} \cdot \frac{4(x+3)}{(x-4)(x+1)}
$$

Cancel:
- $(x+1)$
- $(x+3)$
- $4$ and $2$: $4/2 = 2$

Left with:
$$
\frac{(x-6)}{1} \cdot \frac{2}{(x-4)} = \frac{2(x-6)}{x-4}
$$

✔ Answer: $\boxed{\frac{2(x-6)}{x-4}}$, $x \ne -3, -1, 4$

---

#### 11. $\frac{25xy^3}{35x^4y^2} \cdot \frac{14xy}{10x^2y^3}$

Simplify each fraction first:

- $\frac{25xy^3}{35x^4y^2} = \frac{25}{35} \cdot \frac{x}{x^4} \cdot \frac{y^3}{y^2} = \frac{5}{7} \cdot x^{-3} \cdot y = \frac{5y}{7x^3}$
- $\frac{14xy}{10x^2y^3} = \frac{14}{10} \cdot \frac{x}{x^2} \cdot \frac{y}{y^3} = \frac{7}{5} \cdot x^{-1} \cdot y^{-2} = \frac{7}{5x y^2}$

Now multiply:
$$
\frac{5y}{7x^3} \cdot \frac{7}{5x y^2} = \frac{5y \cdot 7}{7x^3 \cdot 5x y^2} = \frac{35y}{35x^4 y^2} = \frac{1}{x^4 y}
$$

✔ Answer: $\boxed{\frac{1}{x^4 y}}$, $x \ne 0, y \ne 0$

---

#### 12. $\frac{4x}{x+1} \cdot \frac{x^2 - 6x - 7}{x^2 - 7x}$

Factor:

- $x^2 - 6x - 7 = (x-7)(x+1)$
- $x^2 - 7x = x(x - 7)$

Now:
$$
\frac{4x}{x+1} \cdot \frac{(x-7)(x+1)}{x(x-7)}
$$

Cancel:
- $x$ (numerator and denominator)
- $(x-7)$
- $(x+1)$

Left with: $4$

✔ Answer: $\boxed{4}$, $x \ne -1, 0, 7$

---

#### 13. $\frac{6x - 30}{x^2 - 7x + 10} \cdot \frac{7x - 14}{6x}$

Factor:

- $6x - 30 = 6(x - 5)$
- $x^2 - 7x + 10 = (x-2)(x-5)$
- $7x - 14 = 7(x - 2)$
- $6x = 6x$

Now:
$$
\frac{6(x-5)}{(x-2)(x-5)} \cdot \frac{7(x-2)}{6x}
$$

Cancel:
- $(x-5)$
- $(x-2)$
- $6$ in numerator and denominator

Left with:
$$
\frac{1}{1} \cdot \frac{7}{x} = \frac{7}{x}
$$

✔ Answer: $\boxed{\frac{7}{x}}$, $x \ne 0, 2, 5$

---

✔ Final Answers Summary:

1. $\boxed{\frac{2x^2}{3}}$
2. $\boxed{\frac{x+2}{3}}$
3. $\boxed{\frac{x-4}{x+5}}$
4. $\boxed{\frac{1}{x-2}}$
5. $\boxed{\frac{1}{3}}$
6. $\boxed{\frac{3}{5x^3}}$
7. $\boxed{\frac{10}{(x-4)(x-2)}}$
8. $\boxed{\frac{9x}{4}}$
9. $\boxed{\frac{15}{2(x+4)}}$
10. $\boxed{\frac{2(x-6)}{x-4}}$
11. $\boxed{\frac{1}{x^4 y}}$
12. $\boxed{4}$
13. $\boxed{\frac{7}{x}}$

Let me know if you'd like these explained further!