Solving Expression with Cube Roots of Unity

The question asks us to evaluate the expression $$(1 + \omega) (1 + \omega^2) (1 + \omega^3) (1 + \omega + \omega^2)$$ where \(1, \omega, \omega^2\) are the cube roots of unity.

Understanding Cube Roots of Unity

The cube roots of unity, \(1, \omega, \omega^2\), have several important properties:

• They are the solutions to the equation $$z^3 = 1$$.
• The sum of the cube roots of unity is always zero: $$1 + \omega + \omega^2 = 0$$.
• The product of the cube roots of unity is one: $$1 \cdot \omega \cdot \omega^2 = \omega^3 = 1$$.
• Higher powers of \(\omega\) can be reduced using \(\omega^3 = 1\). For example, \(\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\). In general, $$\omega^n = \omega^{n \pmod 3}$$.

Evaluating Each Term of the Expression

Let's look at each part of the given expression: $$(1 + \omega) (1 + \omega^2) (1 + \omega^3) (1 + \omega + \omega^2)$$

We will evaluate each factor separately using the properties of cube roots of unity.

Calculating the Final Value

Now we substitute these evaluated values back into the original expression:

$$(1 + \omega) (1 + \omega^2) (1 + \omega^3) (1 + \omega + \omega^2) = (-\omega^2) \cdot (-\omega) \cdot (2) \cdot (0)$$

When any number or expression is multiplied by zero, the result is always zero.

So, the value of the expression is:

$$-\omega^2 \cdot -\omega \cdot 2 \cdot 0 = 0$$

Conclusion on Cube Roots Expression

The value of the expression \((1 + \omega) (1 + \omega^2) (1 + \omega^3) (1 + \omega + \omega^2)\) is 0.

Revision Table: Cube Roots of Unity Properties

Additional Information on Complex Numbers and Cube Roots

The cube roots of unity are complex numbers that lie on the unit circle in the complex plane. Their values are:

• $$1$$ (which is $$e^{i 0}$$)
• $$\omega = e^{i 2\pi/3} = \cos(2\pi/3) + i \sin(2\pi/3) = -\frac{1}{2} + i \frac{\sqrt{3}}{2}$$
• $$\omega^2 = e^{i 4\pi/3} = \cos(4\pi/3) + i \sin(4\pi/3) = -\frac{1}{2} - i \frac{\sqrt{3}}{2}$$

These roots form an equilateral triangle inscribed in the unit circle centered at the origin. The properties \(1 + \omega + \omega^2 = 0\) and \(\omega^3 = 1\) are fundamental when working with these roots in various mathematical problems.