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

Answer

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

### Step 1: Rewrite the expression
We can factor out the negative sign from the expression:
\[ -\left(\frac{1}{2} - \frac{\sqrt{3}}{2} i\right)^2 = -\left(\frac{1}{2} + \left(-\frac{\sqrt{3}}{2}\right) i\right)^2 \]

### Step 2: Identify \(a\) and \(b\)
Let:
\[ a = \frac{1}{2}, \quad b = -\frac{\sqrt{3}}{2} i \]

### Step 3: Apply the formula for \((a + b)^2\)
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ \left(\frac{1}{2} - \frac{\sqrt{3}}{2} i\right)^2 = a^2 + 2ab + b^2 \]
Calculating each term:
- \(a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(b^2 = \left(-\frac{\sqrt{3}}{2} i\right)^2 = \left(-\frac{\sqrt{3}}{2}\right)^2 i^2 = \frac{3}{4}(-1) = -\frac{3}{4}\)
- \(2ab = 2 \cdot \frac{1}{2} \cdot \left(-\frac{\sqrt{3}}{2} i\right) = -\frac{\sqrt{3}}{2} i\)

### Step 4: Combine the results
Now, substituting back into the equation:
\[ \left(\frac{1}{2} - \frac{\sqrt{3}}{2} i\right)^2 = \frac{1}{4} - \frac{3}{4} - \frac{\sqrt{3}}{2} i \]
This simplifies to:
\[ \frac{1}{4} - \frac{3}{4} = -\frac{2}{4} = -\frac{1}{2} \]
Thus, we have:
\[ \left(\frac{1}{2} - \frac{\sqrt{3}}{2} i\right)^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \]

### Step 5: Apply the negative sign
Now, applying the negative sign we factored out in Step 1:
\[ -\left(-\frac{1}{2} - \frac{\sqrt{3}}{2} i\right) = \frac{1}{2} + \frac{\sqrt{3}}{2} i \]

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