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.