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The image you uploaded contains two sets of exercises related to rational expressions. Let's go through the instructions and some examples to help you understand the solutions:

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### **First Section**: 
_“Give the common factors of the numerator and the denominator of the following rational expressions.”_

1. **\( \frac{4c}{6c} \)**  
   - Common factor: \( c \).
   
2. **\( \frac{2xy}{3x} \)**  
   - Common factor: \( x \).

3. **\( \frac{8a^2c}{-4ac^2} \)**  
   - Common factor: \( 4ac \).

4. **\( \frac{-14abc}{12ac} \)**  
   - Common factor: \( 2ac \).

5. **\( \frac{(-3r)(-4s)}{-18r^2} \)**  
   - Common factor: \( 3r \).

6. **\( \frac{(3m^2)(-6r)}{(-4r^2)(9m)} \)**  
   - Common factor: \( 3mr \).

7. **\( \frac{3x - 12}{16 - 4x} \)**  
   - Common factor: \( 4 \) (in both numerator and denominator).

8. **\( \frac{5x - 15}{9 - 3x} \)**  
   - Common factor: \( 5 \) (numerator) and \( 3 \) (denominator).

9. **\( \frac{x^2 - 9}{(3 - x)^2} \)**  
   - Common factor: \( (x - 3)(x + 3) \).

10. **\( \frac{(a - b)(b - c)}{(a^2 - b^2)(b - c)^2} \)**  
    - Common factor: \( (b - c) \).

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### **Second Section**: 
_“Reduce the given expressions to simplest forms.”_

11. **\( \frac{9m}{-12n} \)**  
    - Simplified form: \( \frac{3m}{-4n} \).

12. **\( \frac{20a}{65b} \)**  
    - Simplified form: \( \frac{4a}{13b} \).

13. **\( \frac{x^2y}{xy} \)**  
    - Simplified form: \( x \).

14. **\( \frac{x^3y^5}{x^4y^2} \)**  
    - Simplified form: \( \frac{y^3}{x} \).

15. **\( \frac{3(x - 2)^2}{6(x - 2)^4} \)**  
    - Simplified form: \( \frac{1}{2(x - 2)^2} \).

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Would you like to continue solving other examples or need any additional explanations? 

Here are five follow-up questions that may help expand your understanding:

1. How do you simplify complex rational expressions with multiple variables?
2. What is the significance of finding common factors in rational expressions?
3. How can you identify and cancel out common factors between the numerator and denominator?
4. Can all rational expressions be simplified, and what makes them irreducible?
5. How does factoring help in reducing rational expressions to their simplest form?

**Tip:** Always factor both the numerator and the denominator completely before canceling out common factors.

Give the common factors of the numerator and the denominator of the following rational expressions, and reduce the given expressions to simplest forms. None of the denominators is zero.

Explore detailed solutions for simplifying and factoring rational expressions suitable for high school students. This guide covers concepts like factoring polynomials, finding the greatest common divisors, and simplifying complex algebraic expressions.

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