If `omega` is a cube root of unity, then
`1+ omega^(2)=` …..

Answer

To solve the problem, we need to find the value of \( 1 + \omega^2 \), where \( \omega \) is a cube root of unity.

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). They are given by:
\[ 1, \quad \omega = e^{2\pi i / 3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2}, \quad \omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \]

2. **Using the Property of Cube Roots of Unity**:
We know that:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can express \( \omega + \omega^2 \) in terms of 1:
\[ \omega + \omega^2 = -1 \]

3. **Finding \( 1 + \omega^2 \)**:
To find \( 1 + \omega^2 \), we can rearrange the equation \( \omega + \omega^2 = -1 \):
\[ 1 + \omega^2 = 1 - \omega \]

4. **Substituting the Value of \( \omega \)**:
Now we substitute the value of \( \omega \):
\[ 1 + \omega^2 = 1 - \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) \]
Simplifying this gives:
\[ 1 + \omega^2 = 1 + \frac{1}{2} - i \frac{\sqrt{3}}{2} = \frac{3}{2} - i \frac{\sqrt{3}}{2} \]

5. **Final Answer**:
Therefore, the value of \( 1 + \omega^2 \) is:
\[ 1 + \omega^2 = \frac{3}{2} - i \frac{\sqrt{3}}{2} \]