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The uploaded image contains a series of algebraic fraction problems that involve simplification of expressions, multiplication, and division of rational expressions. I will solve each problem step by step:

### 1) \(\frac{7(m - 6)}{m - 6} \cdot \frac{5m(7m - 5)}{7(7m - 5)}\)

First, cancel out the common factors:
- \( m - 6 \) appears in both the numerator and denominator, so they cancel out.
- \( 7m - 5 \) also cancels.

What remains is:
\[
\frac{7}{7} \cdot 5m = 5m
\]

### 2) \(\frac{7}{10(n + 3)} \div \frac{n - 2}{(n + 3)(n - 2)}\)

For division, multiply by the reciprocal:
\[
\frac{7}{10(n + 3)} \cdot \frac{(n + 3)(n - 2)}{n - 2}
\]
The \( n - 2 \) terms cancel out, and \( n + 3 \) also cancels. This leaves:
\[
\frac{7}{10}
\]

### 3) \(\frac{7r}{7r(r + 10)} \div \frac{r - 6}{(r - 6)^2}\)

First, simplify the fractions:
\[
\frac{7r}{7r(r + 10)} = \frac{1}{r + 10}
\quad \text{and} \quad
\frac{r - 6}{(r - 6)^2} = \frac{1}{r - 6}
\]
Now, dividing by a fraction is the same as multiplying by the reciprocal:
\[
\frac{1}{r + 10} \cdot \frac{(r - 6)}{1} = \frac{r - 6}{r + 10}
\]

### 4) \(\frac{6x(x + 4)}{x - 3} \cdot \frac{(x - 3)(x - 6)}{6x(x - 6)}\)

Cancel out the common factors \( x - 3 \), \( x - 6 \), and \( 6x \):
\[
\frac{6x(x + 4)}{x - 3} \cdot \frac{(x - 3)(x - 6)}{6x(x - 6)} = 1
\]

### 5) \(\frac{25n + 25}{5} \cdot \frac{4}{30n + 30}\)

Factor the numerators and denominators:
\[
\frac{25n + 25}{5} = \frac{25(n + 1)}{5} = 5(n + 1)
\quad \text{and} \quad
\frac{30n + 30}{30} = \frac{30(n + 1)}{30} = (n + 1)
\]
Now multiply:
\[
5(n + 1) \cdot \frac{4}{30(n + 1)} = \frac{20(n + 1)}{30(n + 1)}
\]
Cancel \( n + 1 \):
\[
\frac{20}{30} = \frac{2}{3}
\]

### 6) \(\frac{9}{b^2 - b - 12} \div \frac{b - 5}{b^2 - b - 12}\)

Factor \( b^2 - b - 12 \):
\[
b^2 - b - 12 = (b - 4)(b + 3)
\]
Now rewrite the expression:
\[
\frac{9}{(b - 4)(b + 3)} \div \frac{b - 5}{(b - 4)(b + 3)}
\]
Cancel the common terms:
\[
\frac{9}{1} \div (b - 5) = \frac{9}{b - 5}
\]

### 7) \(\frac{x - 10}{35x + 21} \div \frac{7}{35x + 21}\)

Simplify the fractions:
\[
\frac{x - 10}{35x + 21} \div \frac{7}{35x + 21} = \frac{x - 10}{7}
\]

### 8) \(\frac{v - 1}{4} \cdot \frac{4}{v^2 - 11v + 10}\)

Factor \( v^2 - 11v + 10 \):
\[
v^2 - 11v + 10 = (v - 1)(v - 10)
\]
Now multiply:
\[
\frac{v - 1}{4} \cdot \frac{4}{(v - 1)(v - 10)} = \frac{1}{v - 10}
\]

### 9) \(\frac{x^2 - 6x - 7}{x + 5} \div \frac{x + 5}{x - 7}\)

Factor the quadratic in the numerator:
\[
x^2 - 6x - 7 = (x - 7)(x + 1)
\]
Now divide:
\[
\frac{(x - 7)(x + 1)}{x + 5} \div \frac{x + 5}{x - 7} = \frac{(x - 7)(x + 1)}{x + 5} \cdot \frac{x - 7}{x + 5}
\]
This simplifies to:
\[
(x - 7)^2 \cdot \frac{x + 1}{(x + 5)^2}
\]

### 10) \(\frac{1}{a - 6} \cdot \frac{8a + 80}{8}\)

Factor \( 8a + 80 \):
\[
8a + 80 = 8(a + 10)
\]
Now multiply:
\[
\frac{1}{a - 6} \cdot \frac{8(a + 10)}{8} = \frac{a + 10}{a - 6}
\]

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These are the step-by-step solutions for each problem.

Would you like further details on any of these solutions?

Here are some related questions to explore further:
1. How do you factor polynomials like in problems 6 and 9?
2. What is the difference between simplifying and solving algebraic fractions?
3. Can you explain how to cancel common factors in a rational expression?
4. Why do we multiply by the reciprocal when dividing fractions?
5. How can you identify common factors quickly in complex rational expressions?

**Tip:** Always check for factoring opportunities before multiplying or dividing rational expressions; this simplifies your work!

A series of algebraic fraction problems involving simplification, multiplication, and division of rational expressions. These problems contain variables such as m, n, r, x, v, and b.

Explore detailed solutions to algebraic fraction problems involving simplification, multiplication, and division of rational expressions. Concepts such as factoring, cancellation of common factors, and polynomial division are covered. Ideal for students in Grades 8-11.

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