https://file.aiquickdraw.com/mathbot/file/1726509423649eabhbkke.jpg

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.

Express z = -3 + j in polar form and hence find z^6, converting your answer into cartesian form.

Learn how to express the complex number z = -3 + j in polar form and use De Moivre's Theorem to find z^6. This problem involves converting polar form back to cartesian form with a detailed, step-by-step solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation