The problem you've uploaded seems to be related to a complex number $z = \frac{(-1 - j)^2}{2}$, where the goal is to represent $z$ in three forms, as well as calculate its powers and roots.

Let's break it down step by step:

Part a) Represent $z$ in three forms (likely Cartesian, Polar, and Exponential form).

Part b) Compute $z^2 - 1$.

Part c) Find the cube roots of $z$.

I will proceed to solve these now. Let's start with part a.

Step 1: Simplify the given expression $z = \frac{(-1 - j)^2}{2}$.

First, let’s simplify $(-1 - j)^2$, where $j$ is the imaginary unit.

$(-1 - j)^2 = (-1)^2 + 2(-1)(-j) + (-j)^2 = 1 + 2j - 1 = 2j$

So, the expression for $z$ becomes: $z = \frac{2j}{2} = j$

Thus, $z = j$, a purely imaginary number.

Part a) Now, we write $z$ in the three standard forms:

1. Cartesian form (a + bi): $z = 0 + 1j$

2. Polar form (r * (cos(θ) + i sin(θ))): The magnitude $r$ is: $r = |z| = |j| = 1$ The argument $\theta$ is $\frac{\pi}{2}$, since $z$ is along the positive imaginary axis.

   Thus, the polar form is: $z = 1 (\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}))$

3. Exponential form (r * e^(iθ)): Using the same values for $r$ and $\theta$: $z = e^{i \frac{\pi}{2}}$

Part b) Now, let's calculate $z^2 - 1$:

We know $z = j$, so: $z^2 = j^2 = -1$ Thus: $z^2 - 1 = -1 - 1 = -2$

Part c) Finding the cube roots of $z$:

To find the cube roots of $z$, we use the polar form of $z = e^{i\frac{\pi}{2}}$.

The cube roots of a complex number are given by: $z_k = r^{1/3} e^{i(\frac{\theta + 2k\pi}{3})}, \quad k = 0, 1, 2$

For $z = e^{i\frac{\pi}{2}}$, $r = 1$ and $\theta = \frac{\pi}{2}$, so the cube roots are: $z_k = e^{i(\frac{\pi/6 + 2k\pi}{3})}, \quad k = 0, 1, 2$

Thus, the cube roots are:

1. $z_0 = e^{i\frac{\pi}{6}}$
2. $z_1 = e^{i\frac{5\pi}{6}}$
3. $z_2 = e^{i\frac{3\pi}{2}}$

These are the three cube roots of $z$.

Would you like a more detailed explanation of any of these steps? Or if you have further questions, feel free to ask!

Here are some related questions:

1. What is the difference between the Cartesian, Polar, and Exponential forms of complex numbers?
2. How can you calculate the roots of a complex number using its polar form?
3. How do you perform complex number exponentiation?
4. What is the significance of the argument in polar and exponential forms?
5. How would you compute higher powers or roots of complex numbers?

Tip: When working with complex numbers in polar or exponential form, always remember to convert between forms carefully, especially when finding powers or roots.