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: $4 \cdot (-1) = -4$

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