https://file.aiquickdraw.com/mathbot/file/1728855710691gv5qz9zk.jpg

The problem in the image asks:

"Čemu je jednako |w| ako je \( w = \frac{\bar{z}}{z^2} \) i \( z = 3i - 3 \)?"

This translates to: 

"What is the modulus \(|w|\) if \( w = \frac{\bar{z}}{z^2} \) and \( z = 3i - 3 \)?"

Let's solve this step-by-step.

1. **Determine \( z \):**  
   We are given \( z = 3i - 3 \). This is a complex number, so we write it as \( z = -3 + 3i \).

2. **Find the conjugate of \( z \):**  
   The conjugate of \( z \), denoted as \( \bar{z} \), is obtained by changing the sign of the imaginary part. Thus,  
   \[
   \bar{z} = -3 - 3i
   \]

3. **Calculate \( z^2 \):**  
   Now, calculate \( z^2 \):
   \[
   z^2 = (-3 + 3i)^2
   \]
   Using the formula \( (a + bi)^2 = a^2 + 2abi + (bi)^2 \), we get:
   \[
   (-3 + 3i)^2 = (-3)^2 + 2(-3)(3i) + (3i)^2 = 9 - 18i + 9i^2
   \]
   Since \( i^2 = -1 \), this simplifies to:
   \[
   z^2 = 9 - 18i - 9 = -18i
   \]

4. **Substitute into \( w \):**  
   Now, substitute \( \bar{z} \) and \( z^2 \) into the expression for \( w \):
   \[
   w = \frac{\bar{z}}{z^2} = \frac{-3 - 3i}{-18i}
   \]
   Simplifying the denominator:
   \[
   w = \frac{-3 - 3i}{-18i} = \frac{-3 - 3i}{-18i} \cdot \frac{-i}{-i} = \frac{(-3 - 3i)(-i)}{-18i^2}
   \]
   Since \( i^2 = -1 \), the denominator becomes:
   \[
   -18i^2 = 18
   \]
   Now simplify the numerator:
   \[
   (-3 - 3i)(-i) = 3i + 3 = 3 + 3i
   \]
   Thus, we have:
   \[
   w = \frac{3 + 3i}{18} = \frac{1}{6} + \frac{1}{6}i
   \]

5. **Find \( |w| \):**  
   The modulus of \( w = \frac{1}{6} + \frac{1}{6}i \) is given by:
   \[
   |w| = \sqrt{\left( \frac{1}{6} \right)^2 + \left( \frac{1}{6} \right)^2} = \sqrt{\frac{1}{36} + \frac{1}{36}} = \sqrt{\frac{2}{36}} = \sqrt{\frac{1}{18}} = \frac{1}{\sqrt{18}} = \frac{1}{3\sqrt{2}} = \frac{\sqrt{2}}{6}
   \]

Thus, the modulus of \( w \), \( |w| \), is \( \frac{\sqrt{2}}{6} \).

---

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions you might find interesting:
1. How do you find the conjugate of a complex number?
2. What is the geometric interpretation of the modulus of a complex number?
3. How can you simplify complex fractions involving imaginary units?
4. What are the properties of the modulus function in complex numbers?
5. How do you square a complex number using the standard form \( a + bi \)?

**Tip:** Always remember that multiplying the numerator and denominator of a fraction by the conjugate is a useful method to simplify expressions with complex numbers.

Čemu je jednako |w| ako je w = \(\frac{\bar{z}}{z^2}\) i z = 3i - 3?

Learn how to find the modulus of the complex number w = \(\frac{\bar{z}}{z^2}\) where z = 3i - 3. This step-by-step solution covers complex number conjugates, modulus calculation, and algebraic manipulation. Suitable for Grades 11-12 students studying pre-calculus or advanced algebra.

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