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Let's break down the given expression step by step.

The sum runs from \( i = 1 \) to \( i = 20 \) for the following expression:

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

### Step-by-step breakdown:

1. **First term inside the parentheses:**
   \[
   \sqrt{\frac{9x^4}{x^8}} \sqrt[4]{i^4}
   \]
   - Simplifying \( \frac{9x^4}{x^8} = \frac{9}{x^4} \).
   - The square root of this expression is \( \sqrt{\frac{9}{x^4}} = \frac{3}{x^2} \).
   - \( \sqrt[4]{i^4} = i \), because raising to the fourth root cancels the exponent of 4.

   So, the first term simplifies to:
   \[
   \frac{3i}{x^2}
   \]

2. **Second term inside the parentheses:**
   \[
   \sqrt{\frac{125}{5}} i^{\log_5 5}
   \]
   - Simplify \( \frac{125}{5} = 25 \), and \( \sqrt{25} = 5 \).
   - \( \log_5 5 = 1 \), so \( i^{\log_5 5} = i^1 = i \).

   Therefore, the second term becomes:
   \[
   5i
   \]

3. **Third term inside the parentheses:**
   \[
   \log_3 243
   \]
   - Since \( 243 = 3^5 \), we have \( \log_3 243 = 5 \).

   This is a constant term.

### Now, putting it all together:

\[
x^2 \left( \frac{3i}{x^2} + 5i + 5 \right)
\]

The \( x^2 \) cancels out with \( \frac{3i}{x^2} \), so the expression simplifies to:

\[
3i + 5i + 5 = 8i + 5
\]

### Final summation:

Now, we need to sum this from \( i = 1 \) to \( i = 20 \):

\[
\sum_{i=1}^{20} (8i + 5)
\]

This can be split into two sums:

\[
8 \sum_{i=1}^{20} i + \sum_{i=1}^{20} 5
\]

- The sum of the first 20 integers is given by the formula \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \), so for \( n = 20 \):

  \[
  \sum_{i=1}^{20} i = \frac{20(21)}{2} = 210
  \]

- The sum of 20 constants (5) is just \( 5 \times 20 = 100 \).

Now calculate the total sum:

\[
8 \times 210 + 100 = 1680 + 100 = 1780
\]

### Final Answer:

The value of the sum is **1780**.

---

Would you like more details or have any questions about the steps? 

Here are five related questions you could explore:
1. How does the sum of an arithmetic sequence work?
2. Can this method be extended to sums with different powers of \( i \)?
3. How do logarithmic functions interact with sums?
4. What is the difference between square roots and fourth roots?
5. How would the result change if the upper limit of the sum was different?

**Tip:** Always simplify the terms inside the sum before performing the summation to avoid complicated expressions.

Find the sum from i = 1 to 20 of the expression involving x^2, roots, logarithms, and powers.

This problem involves finding the sum of a complex expression containing logarithmic, exponential, and root terms. Suitable for advanced high school or college-level students, this example demonstrates the use of summation, logarithmic properties, and arithmetic series.

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