The problem involves expressing the given complex number $(\sqrt{5} - i)^8$ in two forms: rectangular form $(x + yi)$ and exponential form $(re^{i\theta})$. Here's how we solve it:

Step 1: Express $\sqrt{5} - i$ in polar form

1. The magnitude $r$ of the complex number is calculated as: $r = \sqrt{(\sqrt{5})^2 + (-1)^2} = \sqrt{5 + 1} = \sqrt{6}.$

2. The argument $\theta$ (angle in radians) is calculated using: $\theta = \tan^{-1}\left(\frac{-1}{\sqrt{5}}\right).$ Since the complex number is in the fourth quadrant, the angle is negative: $\theta = -\tan^{-1}\left(\frac{1}{\sqrt{5}}\right).$

   Let $\theta_0 = \tan^{-1}\left(\frac{1}{\sqrt{5}}\right)$, so $\theta = -\theta_0$.

   Using a calculator: $\tan^{-1}\left(\frac{1}{\sqrt{5}}\right) \approx 0.4049 \, \text{radians.}$ Thus: $\theta \approx -0.4049.$

So, the polar form of $\sqrt{5} - i$ is: $\sqrt{6} \, e^{-i0.4049}.$

Step 2: Raise the number to the power of 8

Using De Moivre's theorem: $\left(r e^{i\theta}\right)^n = r^n e^{in\theta}.$ Here, $r = \sqrt{6}$, $\theta = -0.4049$, and $n = 8$. Compute:

1. $r^8 = (\sqrt{6})^8 = (6^4) = 1296.$
2. $n\theta = 8 \cdot (-0.4049) = -3.2392 \, \text{radians.}$

Thus, the result in exponential form is: $1296 \, e^{-i3.2392}.$

Step 3: Convert back to rectangular form

To convert to rectangular form, use: $x + yi = r \cos(\theta) + i r \sin(\theta),$ where $r = 1296$ and $\theta = -3.2392$. Compute:

1. $\cos(-3.2392) \approx -0.9950$,
2. $\sin(-3.2392) \approx -0.0995$.

So: $x + yi = 1296 \cdot (-0.9950) + i \cdot 1296 \cdot (-0.0995),$ $x + yi \approx -1289.52 - 128.95i.$

Final Answer:

1. Rectangular form: $-1289.52 - 128.95i$,
2. Exponential form: $1296 \, e^{-i3.2392}$.

Would you like a deeper explanation of any step or clarification? Here are some related questions you might find helpful:

1. How do you derive the formula for converting a complex number to polar form?
2. What is De Moivre's theorem, and why is it useful for powers of complex numbers?
3. How do you calculate $\tan^{-1}$ values manually or using a calculator?
4. Why does the quadrant of the angle affect the value of $\theta$?
5. What are the advantages of polar and exponential forms over rectangular form in complex calculations?

Tip: Always confirm the quadrant of the angle when converting to polar form—it determines the sign of the argument!