To solve the problem, let’s go through each part step by step.

Problem Breakdown:

The complex number $z$ is given by: $z = -\frac{1}{2} - \frac{\sqrt{3}}{2} i$

We need to determine the power $z^5$ using polar coordinates.

Step 1: Convert $z$ to Polar Form

To express $z$ in polar form, we need its modulus $r$ and argument $\varphi$.

Modulus $r$:

$r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$

Thus, $r = 1$.

Argument $\varphi$:

Since $z = -\frac{1}{2} - \frac{\sqrt{3}}{2} i$, it is in the third quadrant of the complex plane.

To find the angle, we calculate the reference angle: $\varphi = \tan^{-1} \left( \frac{\sqrt{3}/2}{1/2} \right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

However, since $z$ is in the third quadrant, the angle $\varphi$ is: $\varphi = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$

So, the polar form of $z$ is: $z = 1 \cdot \left( \cos\frac{4\pi}{3} + i \sin\frac{4\pi}{3} \right)$

Step 2: Calculate $z^5$ Using De Moivre’s Theorem

De Moivre’s Theorem states: $z^n = r^n \left( \cos(n\varphi) + i \sin(n\varphi) \right)$

For $z^5$:

1. $r^5 = 1^5 = 1$
2. $5 \cdot \varphi = 5 \cdot \frac{4\pi}{3} = \frac{20\pi}{3}$

To reduce $\frac{20\pi}{3}$ to the interval $[0, 2\pi)$: $\frac{20\pi}{3} = 2\pi + \frac{14\pi}{3} = 4\pi + \frac{2\pi}{3} = \frac{2\pi}{3}$

So, $z^5 = 1 \cdot \left( \cos\frac{2\pi}{3} + i \sin\frac{2\pi}{3} \right)$

Step 3: Evaluate $\cos\frac{2\pi}{3}$ and $\sin\frac{2\pi}{3}$

$\cos\frac{2\pi}{3} = -\frac{1}{2}, \quad \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}$

Thus, $z^5 = -\frac{1}{2} + \frac{\sqrt{3}}{2} i$

Final Answer:

$z^5 = -\frac{1}{2} + \frac{\sqrt{3}}{2} i$

Let me know if you need further details or have questions.

Here are some related questions for deeper understanding:

1. How would you find $z^n$ for other values of $n$?
2. What is De Moivre's Theorem and how is it used in complex numbers?
3. How do polar and rectangular forms of complex numbers relate?
4. How can you determine which quadrant an angle falls in using trigonometric values?
5. What are other applications of complex number powers in physics and engineering?

Tip: When working with complex numbers in polar form, always ensure the argument angle is within the correct interval, typically $[0, 2\pi)$ or $[-\pi, \pi)$, for consistency.