if z = 3 + 2i and w = 5-2i, determine:
Re(z)
z + w
z * w

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.

Explore how to solve problems involving complex numbers z = 3 + 2i and w = 5 - 2i. Learn to find Re(z), calculate z + w, and z * w with detailed step-by-step solutions. Suitable for advanced high school students.

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