Write the complex number in a + ib form using cube roots of unity: `(-(1)/(2) + sqrt(3)/(2)i)^(1000)`

Explanation

To solve the problem of writing the complex number \((- \frac{1}{2} + \frac{\sqrt{3}}{2} i)^{1000}\) in the form \(a + ib\) using cube roots of unity, we can follow these steps:

### Step 1: Identify the complex number as a cube root of unity
Let \(\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i\). This is one of the cube roots of unity. We know that the cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
and \(\omega^3 = 1\).

### Step 2: Express the power in terms of \(\omega\)
We need to compute \(\omega^{1000}\). Since \(\omega^3 = 1\), we can reduce the exponent modulo 3:
\[ 1000 \mod 3 = 1 \]
This means:
\[ \omega^{1000} = \omega^{3 \cdot 333 + 1} = (\omega^3)^{333} \cdot \omega^1 = 1^{333} \cdot \omega = \omega \]

### Step 3: Write \(\omega\) in \(a + ib\) form
Now we substitute back for \(\omega\):
\[ \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]
Thus, we can express \(\omega^{1000}\) in the form \(a + ib\):
\[ \omega^{1000} = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]

### Final Answer
The complex number \((- \frac{1}{2} + \frac{\sqrt{3}}{2} i)^{1000}\) in the form \(a + ib\) is:
\[ -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]