We are given the complex number $z = -3 + j$ and need to convert it into polar form and subsequently find $z^6$ in Cartesian form.

Step 1: Polar Form of $z$

We are given the polar form of $z$ as:

$z = \sqrt{10} \angle 2.82 \, \text{radians}$

This represents:

• Modulus $|z| = \sqrt{10}$
• Argument $\theta = 2.82$ radians.

Step 2: Finding $z^6$

To find $z^6$ in polar form, we use De Moivre's Theorem: $z^6 = (|z|^6) \angle (6\theta)$

1. Modulus: $|z^6| = (\sqrt{10})^6 = 10^3 = 1000$.
2. Argument: $6\theta = 6 \times 2.82 = 16.92$ radians.

Since angles greater than $2\pi$ radians must be reduced to within one full revolution, we reduce $16.92$ radians modulo $2\pi$. Using $2\pi \approx 6.283$, we perform:

$16.92 \mod 6.283 \approx 4.354 \, \text{radians}$

Thus, the polar form of $z^6$ is: $z^6 = 1000 \angle 4.354 \, \text{radians}$

Step 3: Converting to Cartesian Form

To convert back to Cartesian form, we use the relations: $x = r \cos(\theta), \quad y = r \sin(\theta)$ where $r = 1000$ and $\theta = 4.354$ radians.

1. $x = 1000 \cos(4.354)$
2. $y = 1000 \sin(4.354)$

Calculating the values:

$x \approx 1000 \times (-0.344) = -344$ $y \approx 1000 \times (-0.939) = -939$

Thus, the Cartesian form of $z^6$ is approximately: $z^6 \approx -344 - 939j$

Final Answer

The Cartesian form of $z^6$ is $\boxed{-344 - 939j}$.

Would you like further details or clarifications? Here are 5 related questions to consider:

1. How is De Moivre's Theorem used to simplify powers of complex numbers?
2. Why do we reduce angles modulo $2\pi$ in polar form?
3. What are the advantages of expressing complex numbers in polar form?
4. How can you convert a Cartesian complex number back into polar form?
5. What is the geometric interpretation of multiplying a complex number by a scalar in polar form?

Tip: Always simplify the argument angle in polar form to a value within $[0, 2\pi]$ when converting back to Cartesian form.