If ` omega ` complex cube root of unity, then `((1 + omega )/(omega ^(2)))^(3)` =

Options

A. 1
B. `-1`
C. `omega`
D. `omega^(2)`

Answer

To solve the problem, we need to evaluate the expression \(\left(\frac{1 + \omega}{\omega^2}\right)^3\), where \(\omega\) is a complex cube root of unity.

### Step-by-Step Solution:

1. **Understanding the Roots of Unity**:
The complex cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]
\[ \omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \]
and \(1 + \omega + \omega^2 = 0\).

2. **Finding \(1 + \omega\)**:
From the property of cube roots of unity:
\[ 1 + \omega + \omega^2 = 0 \implies 1 + \omega = -\omega^2 \]

3. **Substituting \(1 + \omega\) into the Expression**:
Now we substitute \(1 + \omega\) into our original expression:
\[ \frac{1 + \omega}{\omega^2} = \frac{-\omega^2}{\omega^2} = -1 \]

4. **Raising to the Power of 3**:
Now we raise the result to the power of 3:
\[ \left(-1\right)^3 = -1 \]

5. **Final Result**:
Therefore, the value of the expression \(\left(\frac{1 + \omega}{\omega^2}\right)^3\) is:
\[ \boxed{-1} \]