https://file.aiquickdraw.com/mathbot/file/1734187505759epmeogzo.jpg

Here is the analysis of the problems in your image, broken into two parts for clarity:

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### **First Set (Simplify Rational Expressions)**
1. **a)** \(\frac{x-3}{3} - \frac{x+5}{4}\)  
   Combine terms by finding the least common denominator (LCD).

2. **b)** \(\frac{2x-4}{9} - \frac{x-3}{4} + 1\)  
   Combine the fractions with LCD, and simplify the result.

3. **c)** \(1 + \frac{3x-4}{4} - \frac{x+2}{3}\)  
   Rewrite \(1\) as a fraction with LCD and simplify step-by-step.

4. **d)** \(\frac{11}{a+11} + \frac{8}{a-8}\)  
   Combine terms using the LCD \((a+11)(a-8)\).

5. **e)** \(\frac{12}{x-12} - \frac{6}{x-6}\)  
   Recognize any relationships between the denominators and simplify.

6. **f)** \(\frac{12}{r+12} + \frac{8}{r-8}\)  
   Find the LCD and simplify.

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### **Second Set (Simplify Exponentials and Algebraic Expressions)**
1. \(16^0\) = **1** (any nonzero number raised to \(0\) is \(1\)).

2. \(16^0\) = **1** (again, same reasoning).

3. \(11^{3x} \times 11^{2x} = 11^{3x+2x} = 11^{5x}\)  
   (Combine exponents by addition).

4. \(10^{4x} \times 10^{2x} = 10^{6x}\).

5. \((6c)^3 = 6^3c^3 = 216c^3\).

6. \((5n)^3 = 5^3n^3 = 125n^3\).

7. \(\frac{2^{-2}}{3^2} = \frac{1}{2^2 \cdot 3^2} = \frac{1}{4 \cdot 9} = \frac{1}{36}\).

8. \(\frac{5}{2^{-3}} = 5 \cdot 2^3 = 5 \cdot 8 = 40\).

9. \(\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}\).

10. \(\frac{a^2}{a^{-1}} = a^{2+1} = a^3\).

11. \(\frac{x^{1/3}y^{-4}}{x^{1/4}y^2} = x^{1/3 - 1/4}y^{-4-2}\).  
    Simplify exponents step-by-step.

12. \(x^{2x+3x+1} = x^{5x+1}\).

13. \(3 \times 3^{2a} \times 3^2 = 3^{1 + 2a + 2} = 3^{2a+3}\).

14. \(\frac{2^m}{2^{-20}} = 2^{m+20}\).

15. \(\frac{2^{x+4}}{2^{x+3}} = 2^{(x+4) - (x+3)} = 2^1 = 2\).

16. \((2a^4)(3ab^2) = 6a^{4+1}b^2 = 6a^5b^2\).

17. \((7m^4n)(8m^6n^5) = 56m^{4+6}n^{1+5} = 56m^{10}n^6\).

18. \(2(-a^7b^5)(-4a^2b^3)(-9a^0b^2)\):  
    Combine coefficients and exponents step-by-step.

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Would you like me to simplify any specific part in detail? Let me know!

### **Related Questions**:
1. How do you find the least common denominator (LCD) for fractions?  
2. What are the rules for combining exponents when multiplying terms?  
3. How do you simplify expressions with negative exponents?  
4. How do you simplify a complex fraction?  
5. How do you expand and simplify algebraic products like \((2a^3)(3b^2)\)?

### **Tip**:  
When simplifying fractions or algebraic expressions, always focus on combining like terms and carefully applying exponent rules.

Simplify the given algebraic expressions and exponents from the image, which include rational expressions, exponentials, and algebraic simplifications.

Explore step-by-step solutions for simplifying rational expressions and exponents, including combining fractions, properties of exponents, and handling negative and zero exponents. Perfect for Grades 8-10 students studying algebra.

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