To evaluate the expression \((2 + 3i)(4 - 5i)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step breakdown of the solution:
### Step 1: Identify the components
We identify the components of the complex numbers:
- \(a = 2\)
- \(b = 3i\)
- \(c = 4\)
- \(d = -5i\)
### Step 2: Apply the distributive property
Using the formula \((a + b)(c + d) = ac + ad + bc + bd\), we can substitute our values:
\[
(2 + 3i)(4 - 5i) = 2 \cdot 4 + 2 \cdot (-5i) + 3i \cdot 4 + 3i \cdot (-5i)
\]
### Step 3: Calculate each term
Now we calculate each term:
1. \(2 \cdot 4 = 8\)
2. \(2 \cdot (-5i) = -10i\)
3. \(3i \cdot 4 = 12i\)
4. \(3i \cdot (-5i) = -15i^2\)
### Step 4: Substitute \(i^2\)
Recall that \(i^2 = -1\). Thus, we can substitute:
\[
-15i^2 = -15(-1) = 15
\]
### Step 5: Combine all terms
Now we combine all the calculated terms:
\[
8 - 10i + 12i + 15
\]
Combine the real parts and the imaginary parts:
\[
(8 + 15) + (-10i + 12i) = 23 + 2i
\]
### Final Answer
Thus, the product of the complex numbers \((2 + 3i)(4 - 5i)\) is:
\[
\boxed{23 + 2i}
\]
To evaluate the expression \((2 + 3i)(4 - 5i)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step breakdown of the solution:
### Step 1: Identify the components
We identify the components of the complex numbers:
- \(a = 2\)
- \(b = 3i\)
- \(c = 4\)
- \(d = -5i\)
...
