Task B-2.1. Given the complex number $z=\frac{(x+i \sqrt{3})^{4}}{1-i}$. Determine all real numbers $x$ for which $|z|=8 \sqrt{2}$.
## Solution.

Let's calculate the modulus of the complex number $z$.

$$
|z|=\left|\frac{(x+i \sqrt{3})^{4}}{1-i}\right|=\frac{|x+i \sqrt{3}|^{4}}{|1-i|}=\frac{\left(\sqrt{x^{2}+3}\right)^{4}}{\sqrt{2}}=\frac{\left(x^{2}+3\right)^{2}}{\sqrt{2}}
$$

Since $|z|=8 \sqrt{2}$, it follows that

$$
\begin{gathered}
\frac{\left(x^{2}+3\right)^{2}}{\sqrt{2}}=8 \sqrt{2} / \cdot \sqrt{2} \\
x^{2}+3= \pm 4 \quad \Rightarrow \quad x^{2}=1 \text { or } x^{2}=-7
\end{gathered}
$$

Since $x$ is a real number, the solutions are $x= \pm 1$.