Convert ` ((1)/(2)+2i)^(3)` in the form of `a+ib`.

Options

A. `-(47)/(2)-(13)/(2)i`
B. `-(47)/(8)-(13)/(2)i`
C. `-(47)/(8)-(13)/(4)i`
D. `-(47)/(4)-(13)/(4)i`

Explanation

To convert \(\left(\frac{1}{2} + 2i\right)^3\) into the form \(a + ib\), we will use the binomial expansion formula for \((a + b)^3\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
Here, \(a = \frac{1}{2}\) and \(b = 2i\).

2. **Apply the binomial expansion**:
The formula for \((a + b)^3\) is:
\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

3. **Calculate each term**:
- \(a^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}\)
- \(b^3 = (2i)^3 = 8i^3 = 8(-i) = -8i\) (since \(i^3 = -i\))
- \(3a^2b = 3\left(\frac{1}{2}\right)^2(2i) = 3 \cdot \frac{1}{4} \cdot 2i = \frac{3}{2}i\)
- \(3ab^2 = 3\left(\frac{1}{2}\right)(2i)^2 = 3 \cdot \frac{1}{2} \cdot 4(-1) = -6\)

4. **Combine all terms**:
Now, we combine all the calculated terms:
\[ \left(\frac{1}{2} + 2i\right)^3 = \frac{1}{8} - 6 + \left(-8i + \frac{3}{2}i\right) \]

5. **Simplify the real part**:
- The real part: \(\frac{1}{8} - 6 = \frac{1}{8} - \frac{48}{8} = -\frac{47}{8}\)

6. **Simplify the imaginary part**:
- The imaginary part: \(-8i + \frac{3}{2}i = -8i + \frac{12}{2}i = -8i + 6i = -2i\)

7. **Final result**:
Therefore, the expression in the form \(a + ib\) is:
\[ -\frac{47}{8} - 2i \]

### Final Answer:
\[ \left(\frac{1}{2} + 2i\right)^3 = -\frac{47}{8} - 2i \]