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Let's break down the given summation and analyze the expression inside the summation step by step. The problem involves finding the sum of the following expression for \( i \) ranging from 1 to 20:

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

### Step 1: Simplifying each part of the expression
1. **First term**: \( x^2 \sqrt{\frac{9x^4}{x^8}} \)
   \[
   \frac{9x^4}{x^8} = \frac{9}{x^4}
   \]
   So,
   \[
   \sqrt{\frac{9}{x^4}} = \frac{3}{x^2}
   \]
   Therefore, the first term simplifies to:
   \[
   x^2 \cdot \frac{3}{x^2} = 3
   \]

2. **Second term**: \( \sqrt{i^4} = i^2 \)
   This simplifies directly to \( i^2 \).

3. **Third term**: \( \sqrt{\frac{125}{5}} \cdot i^{\log_5{5}} \)
   \[
   \frac{125}{5} = 25 \quad \text{and} \quad \sqrt{25} = 5
   \]
   Also, since \( \log_5{5} = 1 \), the term \( i^{\log_5{5}} = i \). Therefore, the third term simplifies to:
   \[
   5i
   \]

4. **Fourth term**: \( \log_3{243} \)
   Since \( 243 = 3^5 \), we have:
   \[
   \log_3{243} = 5
   \]

### Step 2: Writing the simplified expression
After simplifying each term, the expression inside the summation becomes:
\[
3 \cdot i^2 + 5i + 5
\]

### Step 3: Setting up the summation
Now, we need to sum this expression for \( i = 1 \) to \( i = 20 \):
\[
\sum_{i=1}^{20} (3i^2 + 5i + 5)
\]

This can be split into three separate sums:
\[
3 \sum_{i=1}^{20} i^2 + 5 \sum_{i=1}^{20} i + 5 \sum_{i=1}^{20} 1
\]

### Step 4: Using summation formulas
We use standard summation formulas:
1. Sum of squares: \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \)
2. Sum of first \( n \) integers: \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \)
3. Sum of constants: \( \sum_{i=1}^{n} 1 = n \)

For \( n = 20 \), we calculate each sum:
1. \( \sum_{i=1}^{20} i^2 = \frac{20(21)(41)}{6} = 2870 \)
2. \( \sum_{i=1}^{20} i = \frac{20(21)}{2} = 210 \)
3. \( \sum_{i=1}^{20} 1 = 20 \)

### Step 5: Final Calculation
Now substitute these values back into the expression:
\[
3(2870) + 5(210) + 5(20)
\]
\[
= 8610 + 1050 + 100 = 9760
\]

### Final Answer:
The value of the given summation is \( \boxed{9760} \).

Let me know if you'd like further details on any step!

Here are some related questions you can try:
1. What is the sum of \( \sum_{i=1}^{n} i^3 \)?
2. How do you find the sum of cubes for the first \( n \) terms?
3. How would the answer change if the upper limit were 10 instead of 20?
4. What is the general formula for summing \( i^k \) for any integer \( k \)?
5. Can we find a closed form for \( \sum_{i=1}^{n} (ai^2 + bi + c) \) where \( a, b, c \) are constants?

**Tip**: When dealing with sums, always look for patterns or simplifications that apply summation formulas to reduce your workload!

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

Solve the summation of an expression involving powers, logarithms, and roots from i=1 to i=20. This problem applies core algebraic concepts, including summation formulas and logarithmic simplifications. Learn how to simplify and compute complex sums step by step.

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