Evaluating Complex Expression with Cube Root of Unity

This solution explains how to find the value of the expression \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) where \(\omega\) is a complex cube root of unity (\(\omega \neq 1\)).

Understanding Cube Roots of Unity

Complex cube roots of unity are numbers that satisfy the equation \(z^3 = 1\), excluding \(z=1\). The roots are \(1\), \(\omega = e^{i 2\pi/3}\), and \(\omega^2 = e^{i 4\pi/3}\). These roots have important properties:

• \(1 + \omega + \omega^2 = 0\)
• \(\omega^3 = 1\)

From the property \(1 + \omega + \omega^2 = 0\), we can derive other useful relations:

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

Step-by-Step Solution

We need to evaluate the expression \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\). Let's simplify the bases of the powers first.

Simplifying the First Term Base: \((1 + \omega - \omega^2)\)

Using the relation \(1 + \omega = -\omega^2\), we can substitute this into the expression:

Simplifying the Second Term Base: \((1 - \omega + \omega^2)\)

Using the relation \(1 + \omega^2 = -\omega\), we can substitute this into the expression:

Evaluating the Powers

Now, substitute the simplified bases back into the original expression:

Let's evaluate each term separately:

1. \((-2\omega^2)^{100}\)
   • \((-2)^{100} \times (\omega^2)^{100}\)
   • \(2^{100} \times \omega^{2 \times 100}\)
   • \(2^{100} \omega^{200}\)

   To simplify \(\omega^{200}\), we use the property \(\omega^3 = 1\). Divide 200 by 3:

   \( 200 \div 3 = 66 \text{ remainder } 2 \)

   So, \(\omega^{200} = \omega^{3 \times 66 + 2} = (\omega^3)^{66} \omega^2 = 1^{66} \omega^2 = \omega^2\).

   Therefore, the first term is \(2^{100} \omega^2\).

2. \((-2\omega)^{100}\)
   • \((-2)^{100} \times (\omega)^{100}\)
   • \(2^{100} \omega^{100}\)

   To simplify \(\omega^{100}\), we use the property \(\omega^3 = 1\). Divide 100 by 3:

   \( 100 \div 3 = 33 \text{ remainder } 1 \)

   So, \(\omega^{100} = \omega^{3 \times 33 + 1} = (\omega^3)^{33} \omega^1 = 1^{33} \omega = \omega\).

   Therefore, the second term is \(2^{100} \omega\).

Combining the Results

Now, add the simplified terms back together:

Factor out the common term \(2^{100}\):

Using the property \(\omega + \omega^2 = -1\), we get:

Final Answer

The value of the expression \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) is \(-2^{100}\).