The question asks for the value of \((\omega_1 - \omega_2)^2\) where \(\omega_1\) and \(\omega_2\) are the two distinct cube roots of unity different from 1.

Understanding Cube Roots of Unity

The cube roots of unity are the solutions to the equation \(z^3 = 1\). These roots are:

• \(z = 1\)
• \(z = \omega\)
• \(z = \omega^2\)

Here, \(\omega\) and \(\omega^2\) are the two distinct roots that are different from 1. Their values in complex form are often given as:

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

A key property of these roots is that their sum is zero: \(1 + \omega + \omega^2 = 0\). Also, \(\omega^3 = 1\).

Calculating (\(\omega_1 - \omega_2\))² Algebraically

Let's consider \(\omega_1\) and \(\omega_2\) to be the two distinct cube roots of unity different from 1. We can set \(\omega_1 = \omega\) and \(\omega_2 = \omega^2\) (or vice versa, the result will be the same because the expression is squared).

We need to calculate \((\omega - \omega^2)^2\). Let's expand this expression:

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

Using the property \(\omega^3 = 1\), we can simplify the middle term:

\(2(\omega)(\omega^2) = 2\omega^3 = 2(1) = 2\)

The last term simplifies using \(\omega^3 = 1\):

\((\omega^2)^2 = \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\)

Substitute these back into the expanded expression:

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

Rearrange the terms:

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

Now, use the property \(1 + \omega + \omega^2 = 0\). From this, we know that \(\omega + \omega^2 = -1\).

Substitute this into the expression:

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

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

Calculating (\(\omega_1 - \omega_2\))² Using Complex Form

Alternatively, we can use the complex number values for \(\omega\) and \(\omega^2\).

\(\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)

\(\omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\)

First, find the difference \(\omega - \omega^2\):

\(\omega - \omega^2 = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) - \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right)\)

\(\omega - \omega^2 = -\frac{1}{2} + i\frac{\sqrt{3}}{2} + \frac{1}{2} + i\frac{\sqrt{3}}{2}\)

\(\omega - \omega^2 = \left(-\frac{1}{2} + \frac{1}{2}\right) + i\left(\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\right)\)

\(\omega - \omega^2 = 0 + i\left(\frac{2\sqrt{3}}{2}\right)\)

\(\omega - \omega^2 = i\sqrt{3}\)

Now, square the result:

\((\omega - \omega^2)^2 = (i\sqrt{3})^2\)

Using the property \(i^2 = -1\):

\((i\sqrt{3})^2 = i^2 \cdot (\sqrt{3})^2\)

\((i\sqrt{3})^2 = (-1) \cdot (3)\)

\((i\sqrt{3})^2 = -3\)

Both methods give the same result.

Revision Table: Key Properties of Cube Roots of Unity

Additional Information: Roots of Unity

The concept of cube roots of unity is a specific case of the more general concept of \(n\)-th roots of unity. The \(n\)-th roots of unity are the complex numbers \(z\) that satisfy the equation \(z^n = 1\).

These roots can be found using De Moivre's Theorem. The \(n\) distinct \(n\)-th roots of unity are given by:

\(z_k = e^{i\frac{2\pi k}{n}} = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right)\), for \(k = 0, 1, 2, \dots, n-1\).

For \(n=3\), the roots are:

• \(k=0\): \(z_0 = \cos(0) + i\sin(0) = 1\)
• \(k=1\): \(z_1 = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\) (this is \(\omega\))
• \(k=2\): \(z_2 = \cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\) (this is \(\omega^2\))

The properties like the sum of roots being zero (\(1 + \omega + \omega^2 = 0\)) generalize for \(n\)-th roots of unity. If the roots are \(z_0, z_1, \dots, z_{n-1}\), then their sum is \(z_0 + z_1 + \dots + z_{n-1} = 0\) (provided \(n > 1\)).

Understanding these properties is crucial for solving problems involving roots of unity.