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

Explanation

To convert \( \left( \frac{1}{2} + 2i \right)^3 \) into the form \( a + ib \), we can use the binomial theorem. Here’s how to do it step by step:

### Step 1: Identify the components
We have \( z = \frac{1}{2} + 2i \). We need to compute \( z^3 \).

### Step 2: Apply the binomial theorem
Using the binomial theorem, we can expand \( (a + b)^3 \):
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
In our case, let \( a = \frac{1}{2} \) and \( b = 2i \).

### Step 3: Calculate each term
1. Calculate \( a^3 \):
\[ a^3 = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]

2. Calculate \( 3a^2b \):
\[ 3a^2b = 3 \left( \frac{1}{2} \right)^2 (2i) = 3 \cdot \frac{1}{4} \cdot 2i = \frac{3}{2}i \]

3. Calculate \( 3ab^2 \):
\[ 3ab^2 = 3 \left( \frac{1}{2} \right) (2i)^2 = 3 \cdot \frac{1}{2} \cdot 4(-1) = -6 \]

4. Calculate \( b^3 \):
\[ b^3 = (2i)^3 = 8i^3 = 8(-i) = -8i \]

### Step 4: Combine all terms
Now, combine all the calculated terms:
\[ z^3 = a^3 + 3a^2b + 3ab^2 + b^3 = \frac{1}{8} + \frac{3}{2}i - 6 - 8i \]

### Step 5: Simplify the expression
Combine the real parts and the imaginary parts:
1. Real part:
\[ \frac{1}{8} - 6 = \frac{1}{8} - \frac{48}{8} = -\frac{47}{8} \]

2. Imaginary part:
\[ \frac{3}{2}i - 8i = \frac{3}{2}i - \frac{16}{2}i = -\frac{13}{2}i \]

### Final Result
Thus, we can write:
\[ \left( \frac{1}{2} + 2i \right)^3 = -\frac{47}{8} - \frac{13}{2}i \]
This is in the form \( a + ib \) where \( a = -\frac{47}{8} \) and \( b = -\frac{13}{2} \).