If ` z_(1) = 2+ 3i and z_(2) = 5-3i " then " z_(1)z_(2)` is

Options

A. `-9-19i`
B. `-9+19i`
C. ` 19-19`
D. `19 +i9`

Answer

To find the product of the complex numbers \( z_1 = 2 + 3i \) and \( z_2 = 5 - 3i \), we can follow these steps:

### Step 1: Write down the complex numbers
We have:
\[ z_1 = 2 + 3i \]
\[ z_2 = 5 - 3i \]

### Step 2: Multiply the complex numbers
To find \( z_1 z_2 \), we use the distributive property (also known as the FOIL method for binomials):
\[ z_1 z_2 = (2 + 3i)(5 - 3i) \]

### Step 3: Apply the distributive property
Now we will distribute each term in \( z_1 \) by each term in \( z_2 \):
\[ = 2 \cdot 5 + 2 \cdot (-3i) + 3i \cdot 5 + 3i \cdot (-3i) \]
Calculating each term:
1. \( 2 \cdot 5 = 10 \)
2. \( 2 \cdot (-3i) = -6i \)
3. \( 3i \cdot 5 = 15i \)
4. \( 3i \cdot (-3i) = -9i^2 \)

### Step 4: Combine the terms
Now we combine all the terms:
\[ = 10 - 6i + 15i - 9i^2 \]
Combine the imaginary parts:
\[ = 10 + (15i - 6i) - 9i^2 \]
\[ = 10 + 9i - 9i^2 \]

### Step 5: Substitute \( i^2 \) with \(-1\)
Recall that \( i^2 = -1 \):
\[ -9i^2 = -9(-1) = 9 \]
Now substitute this back into the equation:
\[ = 10 + 9i + 9 \]

### Step 6: Simplify the expression
Combine the real parts:
\[ = (10 + 9) + 9i = 19 + 9i \]

### Final Answer
Thus, the product \( z_1 z_2 \) is:
\[ \boxed{19 + 9i} \]