https://file.aiquickdraw.com/mathbot/file/1728855600166zdd93kw3.jpg

From the image, it appears you have an expression involving complex numbers. Here's what I can read from the text:

\[
w = -\frac{z}{z^2}
\]
with the given condition that \( z = 1 - 3i \), where \( i \) represents the imaginary unit.

Let me solve this step by step:

1. **Given equation**: 
   \[
   w = -\frac{z}{z^2}
   \]

2. **Substitute** \( z = 1 - 3i \):
   \[
   w = -\frac{1 - 3i}{(1 - 3i)^2}
   \]

3. **Simplify \( (1 - 3i)^2 \)**:
   \[
   (1 - 3i)^2 = 1^2 - 2(1)(3i) + (3i)^2 = 1 - 6i + 9(-1) = 1 - 6i - 9 = -8 - 6i
   \]

4. **Now substitute this result back into the expression for \( w \)**:
   \[
   w = -\frac{1 - 3i}{-8 - 6i}
   \]

5. **Simplify the fraction** by multiplying both the numerator and denominator by the conjugate of the denominator \( -8 + 6i \):
   \[
   w = -\frac{(1 - 3i)(-8 + 6i)}{(-8 - 6i)(-8 + 6i)}
   \]

6. **Simplify the denominator** \( (-8 - 6i)(-8 + 6i) \):
   \[
   (-8 - 6i)(-8 + 6i) = 64 + 36 = 100
   \]

7. **Expand the numerator** \( (1 - 3i)(-8 + 6i) \):
   \[
   (1)(-8) + (1)(6i) + (-3i)(-8) + (-3i)(6i) = -8 + 6i + 24i - 18i^2 = -8 + 30i + 18 = 10 + 30i
   \]

8. **Thus, the expression for \( w \) becomes**:
   \[
   w = -\frac{10 + 30i}{100} = -\left( \frac{10}{100} + \frac{30i}{100} \right) = -\left( 0.1 + 0.3i \right)
   \]

9. **Final result**:
   \[
   w = -0.1 - 0.3i
   \]

So, \( w = -0.1 - 0.3i \).

Do you want further details or clarifications? 

### Expanding your understanding:
1. How do you calculate the modulus of a complex number like \( w = -0.1 - 0.3i \)?
2. What is the geometric interpretation of multiplying by the conjugate when simplifying complex fractions?
3. Can you find the argument of \( w \)?
4. How does multiplying complex numbers affect their magnitudes and angles in the complex plane?
5. What are other methods to simplify complex fractions?

### Tip:
When simplifying complex expressions, always try to multiply by the conjugate to eliminate the imaginary part in the denominator.

Find the value of w given that w = -z/z^2 and z = 1 - 3i.

Learn how to solve for w in the expression w = -z/z^2 with z = 1 - 3i using complex number properties. This step-by-step guide includes the use of conjugates and fraction simplification, suitable for students in Grades 10-12.

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