If `i=sqrt(-1)`, what is the product of `(4+7i) and ((1)/(2)-2i)`?

Options

A. `16-(9)/(2)i`
B. `14+(9)/(2)i`
C. `2-8i-14i^(2)`
D. `i(8+(9)/(2))`

Answer

To find the product of \( (4 + 7i) \) and \( \left( \frac{1}{2} - 2i \right) \), we can use the distributive property (also known as the FOIL method for binomials). Let's break it down step by step.

### Step 1: Write down the expression
We need to calculate:
\[ (4 + 7i) \left( \frac{1}{2} - 2i \right) \]

### Step 2: Distribute the terms
Now, we will distribute each term in the first expression to each term in the second expression:
\[ = 4 \cdot \frac{1}{2} + 4 \cdot (-2i) + 7i \cdot \frac{1}{2} + 7i \cdot (-2i) \]

### Step 3: Calculate each multiplication
Now we calculate each of these products:
1. \( 4 \cdot \frac{1}{2} = 2 \)
2. \( 4 \cdot (-2i) = -8i \)
3. \( 7i \cdot \frac{1}{2} = \frac{7i}{2} \)
4. \( 7i \cdot (-2i) = -14i^2 \)

### Step 4: Substitute \( i^2 \) with \(-1\)
Recall that \( i^2 = -1 \). Therefore, we can substitute:
\[ -14i^2 = -14(-1) = 14 \]

### Step 5: Combine all the results
Now we combine all the results:
\[ = 2 - 8i + \frac{7i}{2} + 14 \]

### Step 6: Combine like terms
Now, we combine the real parts and the imaginary parts:
- Real parts: \( 2 + 14 = 16 \)
- Imaginary parts: \( -8i + \frac{7i}{2} \)

To combine the imaginary parts, we need a common denominator:
\[ -8i = -\frac{16i}{2} \]
Thus,
\[ -\frac{16i}{2} + \frac{7i}{2} = -\frac{16i - 7i}{2} = -\frac{9i}{2} \]

### Final Result
Putting it all together, we have:
\[ 16 - \frac{9i}{2} \]

Thus, the product of \( (4 + 7i) \) and \( \left( \frac{1}{2} - 2i \right) \) is:
\[ \boxed{16 - \frac{9i}{2}} \]