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не бери логарифм в корень где 125\5, умножай на него а не бери в корень

Let's solve the expression provided in the summation:

The given expression is:

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

Let's break down the expression step by step:

1. **Simplify the first term:**
   \[
   x^2 \sqrt{\frac{9x^4}{x^8}} \cdot \sqrt{i^4}
   \]
   \[
   \sqrt{\frac{9x^4}{x^8}} = \sqrt{\frac{9}{x^4}} = \frac{3}{x^2}
   \]
   \[
   \sqrt{i^4} = i^2
   \]
   So, the first term simplifies to:
   \[
   x^2 \cdot \frac{3}{x^2} \cdot i^2 = 3i^2
   \]

2. **Simplify the second term:**
   \[
   \sqrt{ \frac{125}{5} } \cdot i^{\log_5 5}
   \]
   \[
   \frac{125}{5} = 25
   \]
   \[
   \sqrt{25} = 5
   \]
   \[
   \log_5 5 = 1 \quad \text{so,} \quad i^{\log_5 5} = i^1 = i
   \]
   Thus, the second term simplifies to:
   \[
   5i
   \]

3. **Simplify the third term:**
   \[
   \log_3 243
   \]
   Since \( 243 = 3^5 \), we have:
   \[
   \log_3 243 = \log_3(3^5) = 5
   \]

Putting everything together, the expression inside the summation simplifies to:
\[
3i^2 + 5i + 5
\]

So, the summation we need to evaluate is:
\[
\sum_{i=1}^{20} \left( 3i^2 + 5i + 5 \right)
\]

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

### Evaluate each summation:

1. **Sum of squares:**
   \[
   \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
   \]
   For \( n = 20 \):
   \[
   \sum_{i=1}^{20} i^2 = \frac{20(20+1)(2(20)+1)}{6} = \frac{20 \cdot 21 \cdot 41}{6} = 2870
   \]

2. **Sum of first \( n \) natural numbers:**
   \[
   \sum_{i=1}^{n} i = \frac{n(n+1)}{2}
   \]
   For \( n = 20 \):
   \[
   \sum_{i=1}^{20} i = \frac{20 \cdot 21}{2} = 210
   \]

3. **Sum of constants:**
   \[
   \sum_{i=1}^{20} 5 = 5 \cdot 20 = 100
   \]

Now, substituting these values into the original expression:

\[
3 \cdot 2870 + 5 \cdot 210 + 100 = 8610 + 1050 + 100 = 9760
\]

So, the final answer is \( \boxed{9760} \).

Would you like more details on any of the steps? 

Here are some related questions:
1. What is the general formula for the sum of squares of the first \( n \) natural numbers?
2. How would the summation change if the limits were different?
3. Can you explain the properties of logarithms used in the solution?
4. What if the logarithmic base was different? How would it affect the solution?
5. How do you simplify radical expressions like \( \sqrt{\frac{9x^4}{x^8}} \)?
6. Can you derive the formula for the sum of the first \( n \) natural numbers?
7. What other methods can be used to compute this summation?
8. How does the exponent \( i^2 \) influence the summation?

**Tip:** When solving complex summations, breaking the problem down into smaller parts and evaluating each one separately can simplify the process and reduce the chances of errors.

Learn how to solve a complex summation problem involving logarithms, square roots, and constants step-by-step. This problem covers key mathematical concepts such as summation formulas, properties of logarithms, and simplification of radical expressions. Suitable for high school students.

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