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The table in the exercise shows a list of expressions that represent processing times \( T(n) \) of algorithms for solving problems of size \( n \). The task is to find the dominant term(s) for each expression and specify the Big-O complexity based on that.

Let's go through each expression and determine the dominant term and its corresponding Big-O notation:

1. **Expression**: \( 5 + 0.001n^3 + 0.025n \)
   - **Dominant term**: \( 0.001n^3 \)
   - **Big-O**: \( O(n^3) \)
   
2. **Expression**: \( 500n + 100n^{1.5} + 50n \log_{10} n \)
   - **Dominant term**: \( 100n^{1.5} \)
   - **Big-O**: \( O(n^{1.5}) \)
   
3. **Expression**: \( 0.3n + 5n^{1.5} + 2.5 \cdot n^{1.75} \)
   - **Dominant term**: \( 2.5 \cdot n^{1.75} \)
   - **Big-O**: \( O(n^{1.75}) \)
   
4. **Expression**: \( n^2 \log_2 n + n (\log_2 n)^2 \)
   - **Dominant term**: \( n^2 \log_2 n \)
   - **Big-O**: \( O(n^2 \log n) \)
   
5. **Expression**: \( n \log_3 n + n \log_2 n \)
   - **Dominant term**: \( n \log_2 n \)
   - **Big-O**: \( O(n \log n) \)

6. **Expression**: \( 3 \log_3 n + \log_2 \log_2 \log_2 n \)
   - **Dominant term**: \( 3 \log_3 n \)
   - **Big-O**: \( O(\log n) \)

7. **Expression**: \( 100n + 0.01n^2 \)
   - **Dominant term**: \( 0.01n^2 \)
   - **Big-O**: \( O(n^2) \)

8. **Expression**: \( 0.01n + 100n^2 \)
   - **Dominant term**: \( 100n^2 \)
   - **Big-O**: \( O(n^2) \)

9. **Expression**: \( 2n + n^{0.5} + 0.5n^{1.25} \)
   - **Dominant term**: \( 0.5n^{1.25} \)
   - **Big-O**: \( O(n^{1.25}) \)

10. **Expression**: \( 0.01n \log_2 n + n(\log_2 n)^2 \)
    - **Dominant term**: \( n (\log_2 n)^2 \)
    - **Big-O**: \( O(n (\log n)^2) \)

11. **Expression**: \( 100n \log_3 n + n^3 + 100n \)
    - **Dominant term**: \( n^3 \)
    - **Big-O**: \( O(n^3) \)

12. **Expression**: \( 0.003n \log_3 n + \log_2 \log_2 n \)
    - **Dominant term**: \( 0.003n \log_3 n \)
    - **Big-O**: \( O(n \log n) \)

Would you like more details or have any questions? Here are some follow-up questions to consider:

1. How is the dominant term used to determine Big-O complexity?
2. Why does the constant coefficient not matter in Big-O notation?
3. What is the difference between \( O(n \log n) \) and \( O(n (\log n)^2) \)?
4. Can two algorithms with different Big-O complexities ever have similar performance for small inputs?
5. What is the significance of logarithmic terms like \( \log_2 n \) in algorithm analysis?

**Tip**: When analyzing algorithms, focus on how terms behave as \( n \) grows large, since Big-O notation is concerned with asymptotic behavior.

Assume that each of the expressions below gives the processing time T(n) spent by an algorithm for solving a problem of size n. Select the dominant term(s) having the steepest increase in n and specify the lowest Big-Oh complexity of each algorithm.

Analyze the dominant terms in algorithm time complexities and identify the lowest Big-O complexity for each. This problem involves understanding logarithmic, polynomial, and constant terms, with solutions targeting students studying algorithm complexity.

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