If `1omega,omega^(2)` are the three cube roots of unity, then what is `((a omega^(6)+bomega^(4)+xomega^(2)))/((b+comega^(10)+aomega^(8)))` equal to ?

Options

A. `a/b`
B. b
C. `omega`
D. `omega^(2)`

Explanation

To solve the expression \(\frac{a \omega^{6} + b \omega^{4} + c \omega^{2}}{b + c \omega^{10} + a \omega^{8}}\), we will use the properties of cube roots of unity. The cube roots of unity are \(1\), \(\omega\), and \(\omega^2\), where \(\omega = e^{2\pi i / 3}\) and satisfies the equation \(\omega^3 = 1\).

### Step-by-Step Solution:

1. **Identify the Powers of \(\omega\)**:
Since \(\omega^3 = 1\), we can reduce the powers of \(\omega\) in the expression:
- \(\omega^6 = (\omega^3)^2 = 1^2 = 1\)
- \(\omega^4 = \omega^{3+1} = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\)
- \(\omega^2\) remains \(\omega^2\)
- \(\omega^{10} = \omega^{9+1} = \omega^9 \cdot \omega = 1 \cdot \omega = \omega\)
- \(\omega^8 = \omega^{6+2} = \omega^6 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2\)

2. **Substituting the Reduced Powers**:
Substitute these values back into the expression:
\[ \frac{a(1) + b(\omega) + c(\omega^2)}{b + c(\omega) + a(\omega^2)} \]
This simplifies to:
\[ \frac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} \]

3. **Multiply Numerator and Denominator by \(\omega\)**:
To simplify further, we multiply both the numerator and denominator by \(\omega\):
\[ \frac{\omega(a + b\omega + c\omega^2)}{\omega(b + c\omega + a\omega^2)} \]
This gives:
\[ \frac{a\omega + b\omega^2 + c\omega^3}{b\omega + c\omega^2 + a\omega^3} \]
Since \(\omega^3 = 1\), we can replace \(\omega^3\) with \(1\):
\[ \frac{a\omega + b\omega^2 + c}{b\omega + c\omega^2 + a} \]

4. **Recognizing the Structure**:
The numerator is \(a\omega + b\omega^2 + c\) and the denominator is \(b\omega + c\omega^2 + a\). Notice that both the numerator and denominator have the same form.

5. **Final Simplification**:
Since the numerator and denominator are similar, we can conclude that the value of the expression simplifies to:
\[ \omega \]

### Final Answer:
\[ \frac{a \omega^{6} + b \omega^{4} + c \omega^{2}}{b + c \omega^{10} + a \omega^{8}} = \omega \]