Find the polar form of z = −1 + j√3. Hence, find z^10.

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.

Learn how to find the polar form of the complex number z = -1 + j√3 and calculate z^10 using De Moivre's Theorem. This problem covers key concepts in complex numbers, including magnitude, argument, and polar form. Suitable for advanced high school students and anyone interested in complex number calculations.

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