To find the polar form of the complex number $z = -1 + j\sqrt{3}$, we'll follow these steps:

Step 1: Calculate the magnitude $r$

The magnitude $r$ of a complex number $z = x + jy$ is given by:

$r = \sqrt{x^2 + y^2}$

For $z = -1 + j\sqrt{3}$, we have $x = -1$ and $y = \sqrt{3}$. Therefore:

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

Step 2: Calculate the argument $\theta$

The argument $\theta$ is found using:

$\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Since $x = -1$ and $y = \sqrt{3}$, the argument $\theta$ is:

$\theta = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right)$

This angle lies in the second quadrant (because $x$ is negative and $y$ is positive), so we need to add $\pi$ to the reference angle:

$\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$

Step 3: Write the polar form of $z$

The polar form of the complex number $z = -1 + j\sqrt{3}$ is:

$z = 2 \left( \cos \frac{2\pi}{3} + j\sin \frac{2\pi}{3} \right)$

Step 4: Find $z^{10}$

Using De Moivre's Theorem, which states:

$z^n = r^n \left( \cos(n\theta) + j\sin(n\theta) \right)$

For $z^{10}$:

$z^{10} = 2^{10} \left( \cos\left(10 \times \frac{2\pi}{3}\right) + j\sin\left(10 \times \frac{2\pi}{3}\right) \right)$

Simplifying $2^{10}$ and the angle:

$z^{10} = 1024 \left( \cos\left(\frac{20\pi}{3}\right) + j\sin\left(\frac{20\pi}{3}\right) \right)$

Since $\frac{20\pi}{3}$ is equivalent to $\frac{2\pi}{3}$ after reducing modulo $2\pi$:

$z^{10} = 1024 \left( \cos\frac{2\pi}{3} + j\sin\frac{2\pi}{3} \right)$

Using the known values:

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

Thus:

$z^{10} = 1024 \left(-\frac{1}{2} + j\frac{\sqrt{3}}{2}\right)$

Finally:

$z^{10} = -512 + j512\sqrt{3}$

This is the value of $z^{10}$.

Would you like any further details on this calculation or have any questions?

Related Questions:

1. How do you convert a complex number from polar to rectangular form?
2. What is the geometric interpretation of raising a complex number to a power?
3. How would the calculation change if $z = 1 - j\sqrt{3}$?
4. Can you derive the polar form for any general complex number $z = x + jy$?
5. What are the implications of the polar form when dealing with complex roots?

Tip:

When working with complex numbers in polar form, always reduce the argument (angle) modulo $2\pi$ to simplify your calculations.