Simplify: `((-1)/(2)- (sqrt3)/(2)i)^(2)`

Answer

To simplify the expression \((-1/2 - \sqrt{3}/2 i)^2\), we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ z = \left(-\frac{1}{2} - \frac{\sqrt{3}}{2} i\right)^2 \]

### Step 2: Use the formula for squaring a binomial
Recall that \((a + b)^2 = a^2 + 2ab + b^2\). Here, we can treat \(-\frac{1}{2}\) as \(a\) and \(-\frac{\sqrt{3}}{2} i\) as \(b\):
\[ z = \left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right)\left(-\frac{\sqrt{3}}{2} i\right) + \left(-\frac{\sqrt{3}}{2} i\right)^2 \]

### Step 3: Calculate each term
1. Calculate \(a^2\):
\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]

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

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

### Step 4: Combine the results
Now we can combine all the terms:
\[ z = \frac{1}{4} + \frac{\sqrt{3}}{2} i - \frac{3}{4} \]
Combine the real parts:
\[ z = \left(\frac{1}{4} - \frac{3}{4}\right) + \frac{\sqrt{3}}{2} i = -\frac{2}{4} + \frac{\sqrt{3}}{2} i = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]

### Final Answer
Thus, the simplified form of the expression is:
\[ z = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]