What is `sqrt((1+_(omega)^(2))/(1+_(omega)))` equal to, where `omega` is the cube root of unity ?

Options

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

Explanation

To solve the expression \( \sqrt{\frac{1 + \omega^2}{1 + \omega}} \) where \( \omega \) is a cube root of unity, we can follow these steps:

### Step 1: Understand the properties of \( \omega \)
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). The roots are \( 1, \omega, \) and \( \omega^2 \), where:
- \( \omega = e^{2\pi i / 3} \)
- \( \omega^2 = e^{4\pi i / 3} \)

We also know that:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can express \( 1 + \omega^2 \) in terms of \( \omega \):
\[ 1 + \omega^2 = -\omega \]

### Step 2: Substitute \( 1 + \omega^2 \) in the expression
Now substituting \( 1 + \omega^2 \) into the expression:
\[ \sqrt{\frac{1 + \omega^2}{1 + \omega}} = \sqrt{\frac{-\omega}{1 + \omega}} \]

### Step 3: Express \( 1 + \omega \) in terms of \( \omega^2 \)
Using the same property, we can express \( 1 + \omega \):
\[ 1 + \omega = -\omega^2 \]

### Step 4: Substitute \( 1 + \omega \) in the expression
Now substituting \( 1 + \omega \) into the expression:
\[ \sqrt{\frac{-\omega}{1 + \omega}} = \sqrt{\frac{-\omega}{-\omega^2}} = \sqrt{\frac{\omega}{\omega^2}} \]

### Step 5: Simplify the expression
We know that \( \frac{\omega}{\omega^2} = \frac{1}{\omega} \):
\[ \sqrt{\frac{\omega}{\omega^2}} = \sqrt{\frac{1}{\omega}} = \frac{1}{\sqrt{\omega}} \]

### Step 6: Find \( \sqrt{\omega} \)
The square root of \( \omega \) can be expressed as:
\[ \sqrt{\omega} = e^{\pi i / 3} = \frac{1}{2} + \frac{\sqrt{3}}{2} i \]

### Step 7: Final expression
Thus, we have:
\[ \frac{1}{\sqrt{\omega}} = \frac{1}{e^{\pi i / 3}} = e^{-\pi i / 3} = \frac{1}{2} - \frac{\sqrt{3}}{2} i \]

### Conclusion
The final result is:
\[ \sqrt{\frac{1 + \omega^2}{1 + \omega}} = \omega \]