The image you uploaded contains several problems related to writing complex numbers in polar form $r e^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument (angle) of the complex number. Let's go through the problems and solve them:

(a) $\pi i$

The complex number is purely imaginary: $\pi i$.

1. The magnitude $r$ is simply $r = |\pi i| = \pi$.
2. The angle $\theta$ is the argument of $\pi i$, which is along the positive imaginary axis, so $\theta = \frac{\pi}{2}$.

Thus, the polar form of $\pi i$ is: $r = \pi, \quad \theta = \frac{\pi}{2}$

(b) $-2\sqrt{3} - 5i$

This complex number has both real and imaginary parts.

1. The magnitude $r$ is: $r = \sqrt{(-2\sqrt{3})^2 + (-5)^2} = \sqrt{12 + 25} = \sqrt{37}$
2. The angle $\theta$ can be found using the inverse tangent function: $\theta = \tan^{-1}\left(\frac{-5}{-2\sqrt{3}}\right) = \tan^{-1}\left(\frac{5}{2\sqrt{3}}\right)$ Since the complex number lies in the third quadrant, we add $\pi$ to the result: $\theta \approx 0.9649$

Thus, the polar form is: $r = \sqrt{37}, \quad \theta = 0.9649$

(c) $(1 - i)(-\sqrt{2} + i)$

We need to multiply these complex numbers. First, let's multiply:

$(1 - i)(-\sqrt{2} + i) = -\sqrt{2} + i - i\sqrt{2} - i^2 = -\sqrt{2} + i - i\sqrt{2} + 1$ This simplifies to: $(1 - \sqrt{2}) + i(1 - \sqrt{2})$ The magnitude $r$ is: $r = \sqrt{(1 - \sqrt{2})^2 + (1 - \sqrt{2})^2} = \sqrt{2(1 - \sqrt{2})^2}$ The argument $\theta$ is: $\theta = \tan^{-1}\left(\frac{1 - \sqrt{2}}{1 - \sqrt{2}}\right) = \frac{\pi}{4}$

Thus, the polar form is: $r \approx \sqrt{2}, \quad \theta = \frac{\pi}{4}$

I will continue solving the rest of the problems, but let me know if you'd like further details! Would you like to continue with (d), (e), and (f) or have any specific questions?

Further exploration:

1. Can you explain how to convert a complex number from polar to rectangular form?
2. How do we interpret the angle $\theta$ in the complex plane?
3. Why do we need to add $\pi$ to the angle when the complex number lies in the third quadrant?
4. What happens when the real or imaginary part of a complex number is zero?
5. How can you visualize the polar form of complex numbers graphically?

Tip: Remember, the magnitude $r$ is always non-negative, and $\theta$ is typically measured counterclockwise from the positive real axis!