Understanding Cube Roots of Unity and the Expression

The question asks for the value of a given expression involving $\omega$, where $1, \omega, \omega^2$ are the cube roots of unity. The cube roots of unity have some fundamental properties that are key to solving this problem.

The main properties are:

• The sum of the cube roots of unity is zero: $1 + \omega + \omega^2 = 0$.
• The cube of $\omega$ is one: $\omega^3 = 1$.

These properties help simplify expressions involving higher powers of $\omega$ and sums like $1+\omega$ or $1+\omega^2$.

Simplifying Higher Powers of ω

The given expression is $(1 + \omega^2)(1 + \omega^4)(1 + \omega^8)(1 + \omega^{16})$. Let's simplify the higher powers of $\omega$ using the property $\omega^3 = 1$. When dealing with cube roots of unity, any power of $\omega$ can be reduced by dividing the exponent by 3 and using the remainder as the new exponent, because $\omega^{3k} = (\omega^3)^k = 1^k = 1$.

• For $\omega^4$: $\omega^4 = \omega^{3 \cdot 1 + 1} = (\omega^3)^1 \cdot \omega^1 = 1 \cdot \omega = \omega$.
• For $\omega^8$: $\omega^8 = \omega^{3 \cdot 2 + 2} = (\omega^3)^2 \cdot \omega^2 = 1^2 \cdot \omega^2 = \omega^2$.
• For $\omega^{16}$: $\omega^{16} = \omega^{3 \cdot 5 + 1} = (\omega^3)^5 \cdot \omega^1 = 1^5 \cdot \omega = \omega$.

So, the expression becomes $(1 + \omega^2)(1 + \omega)(1 + \omega^2)(1 + \omega)$.

Using the Sum Property of Cube Roots of Unity

Now, let's use the property $1 + \omega + \omega^2 = 0$ to simplify the terms within the parentheses.

• From $1 + \omega + \omega^2 = 0$, we get $1 + \omega^2 = -\omega$.
• From $1 + \omega + \omega^2 = 0$, we get $1 + \omega = -\omega^2$.

Substituting these into the simplified expression:

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

Evaluating the Final Expression

Now, we multiply the terms:

The expression is $(-\omega)(-\omega^2)(-\omega)(-\omega^2)$.

This can be grouped as $((-\omega)(-\omega^2)) \cdot ((-\omega)(-\omega^2))$.

• $(-\omega)(-\omega^2) = \omega \cdot \omega^2 = \omega^3$.

So the expression becomes $(\omega^3) \cdot (\omega^3)$.

Using the property $\omega^3 = 1$ for the cube roots of unity:

$\omega^3 \cdot \omega^3 = 1 \cdot 1 = 1$.

Thus, the value of the expression $(1 + \omega^2)(1 + \omega^4)(1 + \omega^8)(1 + \omega^{16})$ is 1.

Understanding the properties of cube roots of unity is crucial for solving problems involving $\omega$ and higher powers. Simplifying higher powers and using the sum property $1 + \omega + \omega^2 = 0$ effectively leads to the solution.