Let's break down the given expression step by step.

We have:

$z = 8 \left( \cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right) \right)$

and

$z^* = 8 \left( \cos(45\pi) + i \sin(45\pi) \right)$

1. First, simplify $z$:

The angle $\frac{5\pi}{4}$ is located in the third quadrant of the unit circle, where both sine and cosine are negative.

$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ $\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

Thus, $z$ becomes:

$z = 8 \left( -\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right) \right)$ $z = 8 \times -\frac{\sqrt{2}}{2} + 8i \times -\frac{\sqrt{2}}{2}$ $z = -4\sqrt{2} - 4i\sqrt{2}$

2. Now, simplify $z^*$:

The angle $45\pi$ is coterminal with $\pi$, since $45\pi \mod 2\pi = \pi$.

For $\pi$:

$\cos(\pi) = -1, \quad \sin(\pi) = 0$

Thus, $z^*$ becomes:

$z^* = 8(-1 + i \cdot 0)$ $z^* = -8$

3. Final expression in rectangular form:

$z = -4\sqrt{2} - 4i\sqrt{2}, \quad z^* = -8$

If you want further details, feel free to ask!

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Here are 5 related questions:

1. What is the geometric interpretation of a complex number in polar form?
2. How do you convert between rectangular and polar forms of complex numbers?
3. What are the properties of complex conjugates?
4. Why is $z^*$ based on $45\pi$ simplified to $\pi$?
5. What is the significance of the unit circle when dealing with complex numbers?

Tip: When finding coterminal angles, subtract multiples of $2\pi$ to reduce the angle to one full rotation or less.