It seems like you're mixing some operations involving summation, powers, roots, and complex numbers. Let's break down and interpret the expression.

I'll rewrite your input into clearer mathematical notation:

[ \text{Sum}\left[ 40x^2 , {i, 1, 20}\right] \cdot \frac{\sqrt{9x^4}}{x^8} + \sqrt[4]{i} + (5i + 5) ]

Step-by-Step Breakdown:

1. Summation: The first part seems to indicate a summation over a constant value $40x^2$ from $i = 1$ to $i = 20$. Since the summation is over a constant term: $\sum_{i=1}^{20} 40x^2 = 20 \times 40x^2 = 800x^2$

2. Simplifying the Fraction: The second part is $\frac{\sqrt{9x^4}}{x^8}$. $\sqrt{9x^4} = 3x^2$ So the fraction simplifies to: $\frac{3x^2}{x^8} = \frac{3}{x^6}$

3. Fourth Root of $i$: The next part is $\sqrt[4]{i}$, which refers to the fourth root of the imaginary unit $i$. The fourth roots of $i$ are complex numbers, given by: $\sqrt[4]{i} = e^{i\pi/8} \text{ or another equivalent form in polar coordinates.}$

4. Expression $5i + 5$: This is a simple complex number, $5i + 5$, which can be written as $5(1 + i)$.

Putting It Together:

So, the entire expression becomes: $800x^2 \cdot \frac{3}{x^6} + \sqrt[4]{i} + 5(1 + i)$ Simplifying the first part: $\frac{800x^2 \cdot 3}{x^6} = \frac{2400}{x^4}$

Thus, the final expression is: $\frac{2400}{x^4} + \sqrt[4]{i} + 5(1 + i)$

Further Details:

1. The term $\sqrt[4]{i}$ can be evaluated further for specific values if needed.
2. The expression is simplified in terms of $x$ and involves both real and complex terms.

Do you want further details on the complex roots or more explanations?

Relative Questions:

1. What are the fourth roots of any complex number, including $i$?
2. How do you simplify expressions involving powers of variables?
3. Can the summation of a constant term be generalized for all constants?
4. How are roots of imaginary numbers computed in polar form?
5. What are the applications of summing polynomials?

Tip:

When simplifying complex expressions, always break them down into individual terms, working with one operation at a time!