Let's break down the problem based on the content of the image you uploaded. You are given the following complex numbers in polar or exponential form:

• $z_1 = 1 - i$
• $z_2 = 3 + 4i$
• $z_3 = e^{i\frac{\pi}{2}}$
• $z_4 = 2e^{-i\frac{\pi}{4}}$

The goal is to convert each of these into their Cartesian (rectangular) form, and possibly perform operations between them. Let's solve the individual parts:

a) $z_1 - z_2 + e^{i\pi}$

1. $z_1 = 1 - i$
2. $z_2 = 3 + 4i$
3. $e^{i\pi} = -1$ (this is Euler's identity)

First, calculate $z_1 - z_2$: $z_1 - z_2 = (1 - i) - (3 + 4i) = 1 - i - 3 - 4i = -2 - 5i$

Now, add $e^{i\pi} = -1$: $(-2 - 5i) + (-1) = -3 - 5i$

Thus, the result for part a) is: $z_1 - z_2 + e^{i\pi} = -3 - 5i$

b) $\frac{z_3}{z_4}$

1. $z_3 = e^{i\frac{\pi}{2}}$ which corresponds to $i$ in Cartesian form (since $e^{i\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = i$).
2. $z_4 = 2e^{-i\frac{\pi}{4}}$

Now we compute $\frac{z_3}{z_4}$: $\frac{z_3}{z_4} = \frac{i}{2e^{-i\frac{\pi}{4}}} = \frac{i}{2} e^{i\frac{\pi}{4}}$ Using $e^{i\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}$, the expression becomes: $\frac{i}{2} \left( \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right)$ Distribute $\frac{i}{2}$: