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: $(1+i)^4 = \left(\sqrt{2}\right)^4 \text{cis}\left(4 \cdot \frac{\pi}{4}\right) = 4 \cdot \text{cis}(\pi) = 4(-1) = -4$

• \