Understanding Cube Roots of Unity

The question asks for the value of a specific expression involving the cube roots of unity. Let's first recall the key properties of the cube roots of unity. If 1, ω, and ω² are the cube roots of unity, they satisfy the following fundamental properties:

• The sum of the cube roots of unity is zero: \(1 + \omega + \omega^2 = 0\)
• The cube of either non-real cube root is one: \(\omega^3 = 1\)

From the first property, we can derive useful relationships:

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

The second property, \(\omega^3 = 1\), is crucial for simplifying powers of ω.

Simplifying the Expression

The given expression is \((1 + \omega) (1 + \omega^2) (1 + \omega^4) (1 + \omega^8)\). We need to simplify the terms with powers of ω greater than or equal to 3 using the property \(\omega^3 = 1\).

• For the term \(1 + \omega^4\): Since \(\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\), the term becomes \(1 + \omega\).
• For the term \(1 + \omega^8\): Since \(\omega^8 = \omega^6 \cdot \omega^2 = (\omega^3)^2 \cdot \omega^2 = 1^2 \cdot \omega^2 = \omega^2\), the term becomes \(1 + \omega^2\).

Substituting these simplified terms back into the original expression, we get:

\((1 + \omega) (1 + \omega^2) (1 + \omega) (1 + \omega^2)\)

This expression can be grouped as:

\(((1 + \omega) (1 + \omega^2))^2\)

Evaluating the Simplified Expression

Now we use the property \(1 + \omega + \omega^2 = 0\) to simplify the factors \((1 + \omega)\) and \((1 + \omega^2)\).

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

Substitute these into the grouped expression:

\((-\omega^2)(-\omega)\)

Multiplying these terms:

\((-\omega^2)(-\omega) = \omega^2 \cdot \omega = \omega^3\)

So, the expression inside the parenthesis simplifies to \(\omega^3\).

The entire expression is then \((\omega^3)^2\).

Using the property \(\omega^3 = 1\), we substitute 1 for \(\omega^3\):

\((1)^2 = 1\)

Thus, the value of the given expression \((1 + \omega) (1 + \omega^2) (1 + \omega^4) (1 + \omega^8)\) is 1.

Summary of Steps

1. Identify the properties of cube roots of unity: \(1 + \omega + \omega^2 = 0\) and \(\omega^3 = 1\).
2. Simplify higher powers of \(\omega\) in the expression using \(\omega^3 = 1\). \(\omega^4 = \omega\) and \(\omega^8 = \omega^2\).
3. Rewrite the expression using the simplified terms: \((1 + \omega)(1 + \omega^2)(1 + \omega)(1 + \omega^2) = ((1 + \omega)(1 + \omega^2))^2\).
4. Use the property \(1 + \omega + \omega^2 = 0\) to substitute \(1 + \omega = -\omega^2\) and \(1 + \omega^2 = -\omega\).
5. Evaluate the product inside the parenthesis: \((-\omega^2)(-\omega) = \omega^3\).
6. Evaluate the final expression: \((\omega^3)^2 = (1)^2 = 1\).

Revision Table: Cube Roots of Unity Properties

Additional Information: Non-real Cube Roots

The non-real cube roots of unity, ω and ω², are complex numbers. They can be expressed in terms of the imaginary unit \(i\) and square roots of 3:

• \(\omega = e^{i 2\pi/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
• \(\omega^2 = e^{i 4\pi/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\)

Note that \(\omega^2\) is the complex conjugate of \(\omega\).

While these specific values exist, problems involving expressions like the one in the question are typically solved using the algebraic properties \(1 + \omega + \omega^2 = 0\) and \(\omega^3 = 1\), as this avoids complex number arithmetic and simplifies calculations significantly.