Comprehensive worksheet featuring 22 problems on simplifying algebraic rational expressions through multiplication and division.

Let's solve each of these algebraic simplification problems step by step. The goal is to simplify each expression using the rules of algebra, especially focusing on:

- Factoring
- Canceling common terms in numerators and denominators
- Handling division as multiplication by the reciprocal
- Simplifying rational expressions

We’ll go through each problem one at a time.

---

1)

$$
\frac{3x}{3x(x - 4)} \div \frac{x - 4}{4(x - 4)}
$$

First, simplify the first fraction:
$$
\frac{3x}{3x(x - 4)} = \frac{1}{x - 4}
$$

Now divide:
$$
\frac{1}{x - 4} \div \frac{x - 4}{4(x - 4)} = \frac{1}{x - 4} \cdot \frac{4(x - 4)}{x - 4}
$$

Cancel $x - 4$ in numerator and denominator:
$$
= \frac{1}{1} \cdot \frac{4}{x - 4} = \frac{4}{x - 4}
$$

✔ Answer: $\boxed{\frac{4}{x - 4}}$

---

2)

$$
\frac{x + 1}{(x - 10)(x + 1)} \cdot \frac{(-x + 1)(x - 3)}{x - 1}
$$

Note: $-x + 1 = -(x - 1)$

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

Cancel $x + 1$, and $x - 1$:
$$
= \frac{1}{x - 10} \cdot \frac{-(x - 3)}{1} = -\frac{x - 3}{x - 10}
$$

✔ Answer: $\boxed{-\frac{x - 3}{x - 10}}$

---

3)

$$
\frac{9a^2(a + 2)}{4a} \div \frac{9a^2(a + 2)}{2}
$$

Divide by multiplying by reciprocal:
$$
= \frac{9a^2(a + 2)}{4a} \cdot \frac{2}{9a^2(a + 2)}
$$

Cancel $9a^2(a + 2)$:
$$
= \frac{1}{4a} \cdot 2 = \frac{2}{4a} = \frac{1}{2a}
$$

✔ Answer: $\boxed{\frac{1}{2a}}$

---

4)

$$
\frac{k + 4}{2} \div \frac{(k + 4)(k - 9)}{(k - 6)(k - 9)}
$$

Multiply by reciprocal:
$$
= \frac{k + 4}{2} \cdot \frac{(k - 6)(k - 9)}{(k + 4)(k - 9)}
$$

Cancel $k + 4$ and $k - 9$:
$$
= \frac{1}{2} \cdot (k - 6) = \frac{k - 6}{2}
$$

✔ Answer: $\boxed{\frac{k - 6}{2}}$

---

5)

$$
\frac{3(p + 1)}{p + 1} \cdot \frac{(p - 3)(p + 3)}{3(p + 3)}
$$

Simplify each part:
- $\frac{3(p+1)}{p+1} = 3$
- $\frac{(p - 3)(p + 3)}{3(p + 3)} = \frac{p - 3}{3}$

Now multiply:
$$
3 \cdot \frac{p - 3}{3} = p - 3
$$

✔ Answer: $\boxed{p - 3}$

---

6)

$$
\frac{3(n - 7)}{(n - 7)(n - 6)} \div \frac{n - 9}{(9 - n)(n - 6)}
$$

First, simplify left side:
$$
\frac{3(n - 7)}{(n - 7)(n - 6)} = \frac{3}{n - 6}
$$

Now right side: note that $9 - n = -(n - 9)$

So:
$$
\frac{n - 9}{(9 - n)(n - 6)} = \frac{n - 9}{-(n - 9)(n - 6)} = \frac{1}{-(n - 6)} = -\frac{1}{n - 6}
$$

Now divide:
$$
\frac{3}{n - 6} \div \left(-\frac{1}{n - 6}\right) = \frac{3}{n - 6} \cdot (- (n - 6)) = -3
$$

✔ Answer: $\boxed{-3}$

---

7)

$$
\frac{(x - 2)(x + 6)}{(x - 5)(x + 6)} \cdot \frac{(x - 5)(x - 8)}{8 - x}
$$

Note: $8 - x = -(x - 8)$

So:
$$
= \frac{(x - 2)\cancel{(x + 6)}}{(x - 5)\cancel{(x + 6)}} \cdot \frac{(x - 5)(x - 8)}{-(x - 8)}
$$

Cancel $(x - 5)$ and $(x - 8)$:
$$
= (x - 2) \cdot \frac{1}{-1} = -(x - 2) = -x + 2
$$

✔ Answer: $\boxed{-x + 2}$

---

8)

$$
\frac{8m^2(m + 1)}{5m} \cdot \frac{5m(m - 7)}{(m - 7)(m + 1)}
$$

Simplify step by step:

Left: $\frac{8m^2(m + 1)}{5m} = \frac{8m(m + 1)}{5}$

Right: $\frac{5m(m - 7)}{(m - 7)(m + 1)} = \frac{5m}{m + 1}$

Now multiply:
$$
\frac{8m(m + 1)}{5} \cdot \frac{5m}{m + 1} = 8m \cdot m = 8m^2
$$

(Cancel $5$, $m + 1$)

✔ Answer: $\boxed{8m^2}$

---

9)

$$
\frac{(r + 4)(r + 10)}{(r + 3)(r + 4)} \div \frac{r + 10}{9r^2}
$$

Simplify left:
$$
\frac{(r + 4)(r + 10)}{(r + 3)(r + 4)} = \frac{r + 10}{r + 3}
$$

Now divide:
$$
\frac{r + 10}{r + 3} \div \frac{r + 10}{9r^2} = \frac{r + 10}{r + 3} \cdot \frac{9r^2}{r + 10}
$$

Cancel $r + 10$:
$$
= \frac{1}{r + 3} \cdot 9r^2 = \frac{9r^2}{r + 3}
$$

✔ Answer: $\boxed{\frac{9r^2}{r + 3}}$

---

10)

$$
\frac{x - 3}{x - 1} \div \frac{7(x - 6)}{(x - 1)(x - 6)}
$$

Multiply by reciprocal:
$$
= \frac{x - 3}{x - 1} \cdot \frac{(x - 1)(x - 6)}{7(x - 6)}
$$

Cancel $x - 1$, $x - 6$:
$$
= \frac{x - 3}{1} \cdot \frac{1}{7} = \frac{x - 3}{7}
$$

✔ Answer: $\boxed{\frac{x - 3}{7}}$

---

11)

$$
\frac{4(4 - n)}{6n(n - 4)} \cdot \frac{3n}{(n - 5)(n + 1)}
$$

Note: $4 - n = -(n - 4)$

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

Cancel $n - 4$:
$$
= \frac{-4}{6n} \cdot \frac{3n}{(n - 5)(n + 1)} = \frac{-2}{3n} \cdot \frac{3n}{(n - 5)(n + 1)}
$$

Cancel $3n$:
$$
= -2 \cdot \frac{1}{(n - 5)(n + 1)} = \frac{-2}{(n - 5)(n + 1)}
$$

✔ Answer: $\boxed{\frac{-2}{(n - 5)(n + 1)}}$

---

12)

$$
\frac{10v^2}{v + 10} \cdot \frac{(v + 10)^2}{(v + 4)(v + 10)}
$$

Simplify:
- $v + 10$ appears in both numerator and denominator

$$
= \frac{10v^2}{\cancel{v + 10}} \cdot \frac{(v + 10)^2}{(v + 4)\cancel{(v + 10)}} = 10v^2 \cdot \frac{v + 10}{v + 4}
$$

Wait — actually:

Better:
$$
= \frac{10v^2}{v + 10} \cdot \frac{(v + 10)^2}{(v + 4)(v + 10)} = 10v^2 \cdot \frac{v + 10}{(v + 4)(v + 10)} \cdot \frac{1}{1}
$$

Wait — better to cancel directly:

$$
= \frac{10v^2}{\cancel{v + 10}} \cdot \frac{(v + 10)^2}{(v + 4)\cancel{(v + 10)}} = 10v^2 \cdot \frac{v + 10}{v + 4}
$$

No — we have:
- One $v + 10$ in denominator
- Two in numerator → so one remains

So:
$$
= \frac{10v^2 (v + 10)}{(v + 4)(v + 10)} \text{? No.}
$$

Actually:

$$
\frac{10v^2}{v + 10} \cdot \frac{(v + 10)^2}{(v + 4)(v + 10)} = \frac{10v^2 \cdot (v + 10)^2}{(v + 10) \cdot (v + 4)(v + 10)} = \frac{10v^2 (v + 10)^2}{(v + 10)^2 (v + 4)} = \frac{10v^2}{v + 4}
$$

✔ Answer: $\boxed{\frac{10v^2}{v + 4}}$

---

13)

$$
\frac{1}{(b - 2)(b + 2)} \cdot \frac{(b + 9)(b - 7)}{b - 7}
$$

Cancel $b - 7$:
$$
= \frac{1}{(b - 2)(b + 2)} \cdot (b + 9) = \frac{b + 9}{(b - 2)(b + 2)}
$$

✔ Answer: $\boxed{\frac{b + 9}{(b - 2)(b + 2)}}$

---

14)

$$
\frac{1}{10n(n - 10)} \cdot \frac{10n(n + 6)}{3}
$$

Cancel $10n$:
$$
= \frac{1}{\cancel{10n}(n - 10)} \cdot \frac{\cancel{10n}(n + 6)}{3} = \frac{n + 6}{3(n - 10)}
$$

✔ Answer: $\boxed{\frac{n + 6}{3(n - 10)}}$

---

15)

$$
\frac{(x - 10)(x - 7)}{x - 7} \div \frac{6(10 - x)}{7}
$$

Simplify left:
$$
\frac{(x - 10)(x - 7)}{x - 7} = x - 10
$$

Note: $10 - x = -(x - 10)$

So:
$$
x - 10 \div \frac{6(-(x - 10))}{7} = (x - 10) \cdot \frac{7}{-6(x - 10)} = \frac{7}{-6} = -\frac{7}{6}
$$

✔ Answer: $\boxed{-\frac{7}{6}}$

---

16)

$$
\frac{10k(7k + 2)}{6k} \cdot \frac{4}{4(7k + 2)}
$$

Simplify:
- $\frac{10k}{6k} = \frac{10}{6} = \frac{5}{3}$
- $7k + 2$ cancels
- $4/4 = 1$

So:
$$
= \frac{5}{3} \cdot 1 = \frac{5}{3}
$$

✔ Answer: $\boxed{\frac{5}{3}}$

---

17)

$$
\frac{6 - a}{10a^2} \div \frac{10(a - 6)}{10a^2(a - 5)}
$$

Note: $6 - a = -(a - 6)$

So:
$$
= \frac{-(a - 6)}{10a^2} \div \frac{10(a - 6)}{10a^2(a - 5)} = \frac{-(a - 6)}{10a^2} \cdot \frac{10a^2(a - 5)}{10(a - 6)}
$$

Cancel:
- $a - 6$
- $10a^2$

$$
= -1 \cdot \frac{a - 5}{10} = -\frac{a - 5}{10}
$$

✔ Answer: $\boxed{-\frac{a - 5}{10}}$

---

18)

$$
\frac{9(x + 3)}{3(x - 7)} \div \frac{x + 2}{3(x - 7)}
$$

Simplify left: $\frac{9(x + 3)}{3(x - 7)} = \frac{3(x + 3)}{x - 7}$

Now divide:
$$
\frac{3(x + 3)}{x - 7} \div \frac{x + 2}{3(x - 7)} = \frac{3(x + 3)}{x - 7} \cdot \frac{3(x - 7)}{x + 2}
$$

Cancel $x - 7$:
$$
= 3(x + 3) \cdot \frac{3}{x + 2} = \frac{9(x + 3)}{x + 2}
$$

✔ Answer: $\boxed{\frac{9(x + 3)}{x + 2}}$

---

19)

$$
\frac{7x^2}{(x - 7)(x - 6)} \cdot \frac{9x(x - 6)}{9x}
$$

Simplify second fraction:
$$
\frac{9x(x - 6)}{9x} = x - 6
$$

So:
$$
= \frac{7x^2}{(x - 7)(x - 6)} \cdot (x - 6) = \frac{7x^2}{x - 7}
$$

✔ Answer: $\boxed{\frac{7x^2}{x - 7}}$

---

20)

$$
\frac{5n}{n - 7} \cdot \frac{9(7 - n)}{9(n - 1)}
$$

Note: $7 - n = -(n - 7)$

So:
$$
= \frac{5n}{n - 7} \cdot \frac{9(-(n - 7))}{9(n - 1)} = \frac{5n}{n - 7} \cdot \frac{- (n - 7)}{n - 1}
$$

Cancel $n - 7$:
$$
= 5n \cdot \frac{-1}{n - 1} = -\frac{5n}{n - 1}
$$

✔ Answer: $\boxed{-\frac{5n}{n - 1}}$

---

21)

$$
\frac{10m^2(3m - 4)}{m + 4} \div \frac{10m^2(3m - 4)}{(m + 2)(m - 3)}
$$

Multiply by reciprocal:
$$
= \frac{10m^2(3m - 4)}{m + 4} \cdot \frac{(m + 2)(m - 3)}{10m^2(3m - 4)}
$$

Cancel all common terms:
- $10m^2$, $3m - 4$

$$
= \frac{1}{m + 4} \cdot (m + 2)(m - 3) = \frac{(m + 2)(m - 3)}{m + 4}
$$

✔ Answer: $\boxed{\frac{(m + 2)(m - 3)}{m + 4}}$

---

22)

$$
\frac{(p - 7)(p + 6)}{(p - 3)(p + 6)} \cdot \frac{(p - 3)(p + 7)}{p + 7}
$$

Cancel:
- $p + 6$
- $p - 3$
- $p + 7$

$$
= (p - 7) \cdot 1 = p - 7
$$

✔ Answer: $\boxed{p - 7}$

---

## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{4}{x - 4}$ |
| 2 | $-\frac{x - 3}{x - 10}$ |
| 3 | $\frac{1}{2a}$ |
| 4 | $\frac{k - 6}{2}$ |
| 5 | $p - 3$ |
| 6 | $-3$ |
| 7 | $-x + 2$ |
| 8 | $8m^2$ |
| 9 | $\frac{9r^2}{r + 3}$ |
| 10 | $\frac{x - 3}{7}$ |
| 11 | $\frac{-2}{(n - 5)(n + 1)}$ |
| 12 | $\frac{10v^2}{v + 4}$ |
| 13 | $\frac{b + 9}{(b - 2)(b + 2)}$ |
| 14 | $\frac{n + 6}{3(n - 10)}$ |
| 15 | $-\frac{7}{6}$ |
| 16 | $\frac{5}{3}$ |
| 17 | $-\frac{a - 5}{10}$ |
| 18 | $\frac{9(x + 3)}{x + 2}$ |
| 19 | $\frac{7x^2}{x - 7}$ |
| 20 | $-\frac{5n}{n - 1}$ |
| 21 | $\frac{(m + 2)(m - 3)}{m + 4}$ |
| 22 | $p - 7$ |

Let me know if you'd like this in a printable format or with steps for any specific problem!