https://file.aiquickdraw.com/mathbot/file/1725797923533168c8i5o.jpg

The equation provided in the image is:

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

I will simplify this expression step by step.

### Step 1: Simplify the numerator

1. **For \((1 + i)^4\)**:
   \[
   1 + i \quad \text{can be rewritten in polar form as} \quad \sqrt{2} \text{cis}\left(\frac{\pi}{4}\right)
   \]
   Therefore:
   \[
   (1 + i)^4 = (\sqrt{2})^4 \text{cis}\left(4 \cdot \frac{\pi}{4}\right) = 4 \text{cis}(\pi)
   \]
   Which simplifies to:
   \[
   4 \cdot (-1) = -4
   \]

2. **

Simplify the complex expression: z = \frac{(1 + i)^4 (1 + i \sqrt{3})^5}{2 \cos \frac{\pi}{20} + 2 i \sin \frac{\pi}{20}}

This solution explores how to simplify a complex number expression involving powers of complex numbers, using Euler's formula and De Moivre's theorem. The equation contains trigonometric functions and polar forms of complex numbers.

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