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Thank you for the image. I can see a set of mathematical exercises, including operations with numbers, exponents, and expressions. I'll go through the problems step-by-step to provide solutions and explanations.

### Problems and Solutions:

1. **Problem 1: \( 21 \times 7 = \)**
   - Solution: \( 21 \times 7 = 147 \)

2. **Problem 2: \( 6 \times 15 = \)**
   - Solution: \( 6 \times 15 = 90 \)

3. **Problem 3: \( 3^2 \times 2^5 = \)**
   - Solution: \( 3^2 = 9 \) and \( 2^5 = 32 \). Therefore, \( 9 \times 32 = 288 \)

4. **Problem 4: \( 269 + 4^2 \times 5 = \)**
   - Solution: \( 4^2 = 16 \), \( 16 \times 5 = 80 \), and \( 269 + 80 = 349 \)

5. **Problem 5: \( 9 + 4 \times 5 (2 + 3) + 42 = \)**
   - Solution: \( 2 + 3 = 5 \), \( 5 \times 5 = 25 \), \( 4 \times 25 = 100 \), and \( 9 + 100 + 42 = 151 \)

6. **Problem 6: \( 4^2 \times 7^2 = \)**
   - Solution: \( 4^2 = 16 \), \( 7^2 = 49 \), \( 16 \times 49 = 784 \)

7. **Problem 7: \( 8^2 - 3^2 + (7 + 2) \div 3 = \)**
   - Solution: \( 8^2 = 64 \), \( 3^2 = 9 \), \( 7 + 2 = 9 \), \( 9 \div 3 = 3 \). Then, \( 64 - 9 + 3 = 58 \)

8. **Problem 8: \( 5^3 - (9 + 5)^2 = \)**
   - Solution: \( 5^3 = 125 \), \( 9 + 5 = 14 \), \( 14^2 = 196 \), \( 125 - 196 = -71 \)

9. **Problem 9: Multiplicar los números de las raíces haciendo 150.**
   - This problem is not clear. Please clarify if it involves any specific roots or additional operations.

10. **Problem 10: Resolver las potencias.**
    - \( \frac{9^2}{4^2} = \frac{81}{16} \)
    - \( \frac{15^3}{3^2} = \frac{3375}{9} = 375 \)
    - \( 5^5 - 3^3 = 3125 - 27 = 3098 \)
    - \( 6^3 \times 4^2 = 216 \times 16 = 3456 \)
    - \( 5^{1.5} \times \frac{8}{2} = 11.18 \times 4 = 44.72 \)

Would you like to see detailed steps for any of these problems or have any specific questions?

### Further Questions for Exploration:
1. How can you verify if your solutions are correct?
2. What are some methods for simplifying expressions involving exponents?
3. How do order of operations (PEMDAS/BODMAS) affect the solutions?
4. Can you derive different approaches to solve exponentiation problems?
5. What are the rules for multiplying and dividing powers with the same base?

### Tip:
When solving complex expressions, always follow the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

Solve the following mathematical exercises involving multiplication, exponentiation, and expressions as seen in the provided image.

Explore and solve various mathematical exercises involving multiplication, exponentiation, and expressions. The problems are suitable for students in Grades 6-8 and cover key concepts like arithmetic operations, properties of exponents, and the order of operations.

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Math Problem Analysis

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