Let's break down each part of the problem.

Given:

• $z = 3 + 2i$
• $w = 5 - 2i$

1. Find $\text{Re}(z)$:

The real part of $z$ is the coefficient of the real term in $z = 3 + 2i$. $\text{Re}(z) = 3$

2. Find $z + w$:

To add the complex numbers $z$ and $w$, add their real parts and their imaginary parts separately. $z + w = (3 + 2i) + (5 - 2i)$ $z + w = (3 + 5) + (2i - 2i)$ $z + w = 8 + 0i = 8$

3. Find $z \times w$:

To multiply the complex numbers $z$ and $w$, apply the distributive property (also known as FOIL): $z \times w = (3 + 2i) \times (5 - 2i)$ $z \times w = 3 \times 5 + 3 \times (-2i) + 2i \times 5 + 2i \times (-2i)$ $z \times w = 15 - 6i + 10i - 4i^2$ Since $i^2 = -1$: $z \times w = 15 - 6i + 10i + 4$ $z \times w = 19 + 4i$

Final Answers:

1. $\text{Re}(z) = 3$
2. $z + w = 8$
3. $z \times w = 19 + 4i$

Would you like more details or have any questions? Here are 5 related questions to explore further:

1. What is the imaginary part of $z$?
2. How do you subtract $w$ from $z$?
3. What is the modulus (absolute value) of $z$?
4. How do you find the conjugate of a complex number?
5. How would you divide $z$ by $w$?

Tip: When multiplying complex numbers, remember to use $i^2 = -1$ to simplify the result.