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The given expression is:

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

You are also asked to express the result in binomial form (i.e., in the form \(a + bi\)).

### Let's break this down into steps:

1. **Simplify the base expression**:
    - The term in the numerator: \(\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i \right)^{20}\) represents a complex number raised to a power. This can be done by converting to polar form and then applying De Moivre's Theorem.
    
    - The second factor in the numerator: \((4 - 4\sqrt{3}i)\) is also a complex number.
    
    - The denominator \(\left( \sqrt{2} + \sqrt{2}i \right)^2\) can be simplified by expanding the square.

2. **Convert the complex numbers to polar form**:
    - For the first term \(\frac{\sqrt{3}}{2} + \frac{1}{2}i\), we can calculate the modulus \(r\) and argument \(\theta\).
    
    - Similarly, convert \(4 - 4\sqrt{3}i\) to polar form.
    
3. **Use De Moivre’s Theorem**:
    - Apply De Moivre’s theorem to simplify \(\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i \right)^{20}\).

4. **Simplify the denominator**:
    - Simplify \(\left( \sqrt{2} + \sqrt{2}i \right)^2\) using algebraic expansion.

5. **Combine the terms**:
    - After simplification, express the result in the form \(a + bi\).

Let me proceed with detailed calculations to simplify this expression.The simplified expression in binomial form is:

\[
0 + 2i
\]

Thus, the result of the given expression is purely imaginary, and it simplifies to \(2i\).

Would you like a more detailed breakdown of any step? Here are five related questions you might explore:

1. How do you convert complex numbers to polar form?
2. What is De Moivre’s Theorem, and how is it applied?
3. How can you simplify complex numbers when raised to large powers?
4. Why does the denominator \( \left( \sqrt{2} + \sqrt{2}i \right)^2 \) simplify so easily?
5. What does it mean when a complex number has no real part?

**Tip**: When raising a complex number to a power, converting it to polar form first can significantly simplify the process.

Simplifica la siguiente expresión:
(√3/2 + 1/2i)^20 (4 - 4√3i) / (√2 + √2i)^2

Learn how to simplify complex numbers involving exponentiation and convert the result into binomial form. This detailed problem demonstrates using De Moivre's theorem, polar forms, and algebraic simplification. Suitable for advanced math students.

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