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The problem asks to simplify the expression:

\[
\frac{\left( \frac{\sqrt{3}}{2} + \frac{1}{2} i \right)^{20} \left( 4 - 4 \sqrt{3} i \right)}{\left( \sqrt{2} + \sqrt{2} i \right)^2}
\]

and express the result in binomial form (in the form \(a + bi\), where \(a\) and \(b\) are real numbers).

### Step-by-step breakdown:

1. **Simplify the numerator**: 
   - \(\left( \frac{\sqrt{3}}{2} + \frac{1}{2} i \right)^{20}\) can be interpreted as a complex number in polar form.
   - \(\frac{\sqrt{3}}{2} + \frac{1}{2} i\) is a complex number with a modulus \(r\) and argument \(\theta\). First, we convert this to polar form, then raise it to the 20th power using De Moivre's Theorem.

2. **Simplify the other terms**: 
   - Multiply the term \(4 - 4 \sqrt{3} i\).
   - Simplify the denominator \(\left( \sqrt{2} + \sqrt{2} i \right)^2\) using standard binomial expansion or by converting to polar form as well.

I'll start solving the expression step-by-step. Let me calculate these parts for you.After simplifying the expression, the result is \(2i\).

Therefore, the expression in binomial form is:

\[
0 + 2i
\]

Would you like to go through any specific part of the solution in detail?

Here are 5 related questions you might find useful:
1. How can we convert a complex number into polar form?
2. What is De Moivre's Theorem, and how is it applied here?
3. Why does the modulus and argument play a key role in simplifying powers of complex numbers?
4. How can binomial expansion be used to simplify complex expressions?
5. What are the geometric interpretations of complex numbers in polar and binomial forms?

**Tip**: When dealing with complex numbers raised to high powers, using polar form and De Moivre's Theorem significantly simplifies the calculations.

Simplifica la siguiente expresión: ( (√3/2 + 1/2i)²⁰ (4 - 4√3i) ) / ( (√2 + √2i)² ). Además, expresa la respuesta en forma binómica.

This page details the simplification of a complex number expression using De Moivre's Theorem, converting to polar form, and binomial expansion. It is suited for advanced high school and university level mathematics.

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