https://file.aiquickdraw.com/mathbot/file/1725798003482xoyhv2u7.jpg

The expression in the image can be written as:

\[
z = \frac{(1+i)^4 \cdot (1 + i\sqrt{3})^5}{\left( 2\cos\left(\frac{\pi}{20}\right) + 2i \sin\left(\frac{\pi}{20}\right)\right)^6}
\]

Let's simplify and solve this step by step:

### 1. Numerator
- \((1+i)^4\): First, express \(1+i\) in polar form:
  \[
  1+i = \sqrt{2} \text{cis}\left(\frac{\pi}{4}\right)
  \]
  Using De Moivre's Theorem for powers of complex numbers:
  \[
  (1+i)^4 = \left(\sqrt{2}\right)^4 \text{cis}\left(4 \cdot \frac{\pi}{4}\right) = 4 \cdot \text{cis}(\pi) = 4(-1) = -4
  \]

- \

Simplify and evaluate the expression z = [(1+i)^4 * (1 + i√3)^5] / [(2 cos(π/20) + 2 i sin(π/20))^6].

Learn how to simplify the complex expression z = [(1+i)^4 * (1 + i√3)^5] / [(2 cos(π/20) + 2 i sin(π/20))^6] using De Moivre's theorem and Euler's formula. Step-by-step explanation of polar form transformations and power calculations. Suitable for advanced algebra students.

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