wrong sqrt take only 125/5*i log not in caps

Thank you for the clarification. Let's correct the calculation with the new instructions.

Given the expression:

\[
\sum_{i=1}^{20} \left( x^2 \sqrt{\frac{9x^4}{x^8}} \cdot \sqrt{i^4} + \sqrt{\frac{125}{5} \cdot i^{\log_5 i^5}} + \log_3 243 \right)
\]

Based on the correction:

1. **Simplify \( \sqrt{\frac{9x^4}{x^8}} \):**
   \[
   \sqrt{\frac{9x^4}{x^8}} = \sqrt{\frac{9}{x^4}} = \frac{3}{x^2}
   \]
   Hence, \( x^2 \cdot \frac{3}{x^2} = 3 \).

2. **Simplify \( \sqrt{i^4} \):**
   \[
   \sqrt{i^4} = i^2
   \]
   The first part simplifies to \( 3i^2 \).

3. **Simplify \( \sqrt{\frac{125}{5} \cdot i^{\log_5 i}} \):**
   - Simplify \( \frac{125}{5} = 25 \).
   - The expression inside the square root is now \( 25 \cdot i^{\log_5 i} \).

   Therefore:
   \[
   \sqrt{25 \cdot i^{\log_5 i}} = 5 \sqrt{i^{\log_5 i}}
   \]

   Since \( \sqrt{i^{\log_5 i}} \) simplifies to \( i^{\log_5 i / 2} \), the entire term becomes:
   \[
   5 \cdot i^{\log_5 i / 2}
   \]

4. **Simplify \( \log_3 243 \):**
   \[
   \log_3 243 = \log_3 3^5 = 5
   \]

So, the expression inside the summation becomes:
\[
3i^2 + 5 \cdot i^{\log_5 i / 2} + 5
\]

### Now, find the sum:
\[
\sum_{i=1}^{20} \left( 3i^2 + 5 \cdot i^{\log_5 i / 2} + 5 \right)
\]

This sum can be split into three separate sums:

1. **Sum of \( 3i^2 \):**
   \[
   3 \sum_{i=1}^{20} i^2 = 8610
   \]

2. **Sum of \( 5 \cdot i^{\log_5 i / 2} \):**
   We will calculate this sum numerically.

3. **Sum of the constant term \( 5 \):**
   \[
   5 \times 20 = 100
   \]

### Calculation of \( \sum_{i=1}^{20} 5 \cdot i^{\log_5 i / 2} \):
Let's compute this sum.The sum of \( 5 \cdot i^{\log_5 i / 2} \) from \( i = 1 \) to \( i = 20 \) is approximately 667.61.

### Final Summation:
Adding up all parts:
\[
\sum_{i=1}^{20} \left( 3i^2 + 5 \cdot i^{\log_5 i / 2} + 5 \right) = 8610 + 667.61 + 100 = 9377.61
\]

So, the final value of the summation is approximately **9377.61**.

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions you might find useful:

1. How does \( i^{\log_5 i / 2} \) affect the overall sum compared to \( i^2 \)?
2. Can the sum be evaluated for different logarithmic bases, like \( \log_2 \) or \( \log_{10} \)?
3. What changes if the square root function is replaced with a cube root?
4. How does the sum \( 3i^2 \) compare with the other components in magnitude?
5. What would happen if the summation range was extended beyond 20?
6. Can you find a general formula for similar summations with different powers and coefficients?
7. What does the term \( i^{\log_5 i / 2} \) signify in mathematical analysis?
8. How do such summations relate to series in calculus?

**Tip:** When dealing with logarithms, ensure the base is clear and appropriately applied, as small changes can significantly impact the result.

Learn how to correct a complex mathematical expression involving square roots, logarithms, and summations step-by-step. Detailed explanation and numerical computation provided.

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