Understanding Complex Cube Roots of Unity ($\omega$)

This problem involves the concept of complex cube roots of unity. A complex cube root of unity is a complex number that, when cubed, equals 1. The principal complex cube root of unity is often denoted by $\omega$. The key properties we'll use are:

• $\omega^3 = 1$
• $1 + \omega + \omega^2 = 0$

From the second property, we can derive other useful relations:

• $1 + \omega = -\omega^2$
• $1 + \omega^2 = -\omega$
• $\omega + \omega^2 = -1$

Also, since $\omega^3 = 1$, any higher power of $\omega$ can be simplified by reducing the exponent modulo 3. For example:

• $\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$
• $\omega^5 = \omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2$
• $\omega^6 = (\omega^3)^2 = 1^2 = 1$
• $\omega^8 = \omega^6 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2$

Simplifying the Expression Factors

The expression we need to evaluate is $(1-\omega+\omega^2)(1-\omega^2+\omega^4)(1-\omega^4+\omega^8)$. Let's simplify each factor separately using the properties mentioned above.

Factor 1: $(1-\omega+\omega^2)$

We can rearrange this term using the property $1 + \omega^2 = -\omega$. Substituting this, we get:

$(1 + \omega^2) - \omega = (-\omega) - \omega = -2\omega$

So, the first factor simplifies to $-2\omega$.

Factor 2: $(1-\omega^2+\omega^4)$

First, simplify the power $\omega^4$. As shown earlier, $\omega^4 = \omega$. So the factor becomes $(1-\omega^2+\omega)$. Now, let's rearrange this using the property $1 + \omega = -\omega^2$. Substituting this, we get:

$(1 + \omega) - \omega^2 = (-\omega^2) - \omega^2 = -2\omega^2$

So, the second factor simplifies to $-2\omega^2$.

Factor 3: $(1-\omega^4+\omega^8)$

Simplify the powers first: $\omega^4 = \omega$ and $\omega^8 = \omega^2$. Substituting these, the factor becomes $(1-\omega+\omega^2)$. This is exactly the same as the first factor, which we found simplifies to $-2\omega$. So, the third factor simplifies to $-2\omega$.

Evaluating the Product of Simplified Factors

Now, we need to multiply the simplified forms of the three factors:

Product = (Factor 1) $\times$ (Factor 2) $\times$ (Factor 3)

Product = $(-2\omega) \times (-2\omega^2) \times (-2\omega)$

Let's multiply the constants and the powers of $\omega$ separately:

Product = $(-2 \times -2 \times -2) \times (\omega \times \omega^2 \times \omega)$

Product = $(-8) \times (\omega^{1+2+1})$

Product = $(-8) \times (\omega^4)$

Final Value Calculation

We know that $\omega^4 = \omega$. Substitute this back into the expression for the product:

Product = $-8 \times \omega$

Product = $-8\omega$

Therefore, the value of the expression $(1-\omega+\omega^2)(1-\omega^2+\omega^4)(1-\omega^4+\omega^8)$ is $-8\omega$.