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:
\( (1 + \omega) - \omega^2 = (-\omega^2) - \omega^2 \) \( = -2\omega^2 \)Simplifying the Second Term Base: \((1 - \omega + \omega^2)\)
Using the relation \(1 + \omega^2 = -\omega\), we can substitute this into the expression:
\( (1 + \omega^2) - \omega = (-\omega) - \omega \) \( = -2\omega \)Evaluating the Powers
Now, substitute the simplified bases back into the original expression:
\( (-2\omega^2)^{100} + (-2\omega)^{100} \)Let's evaluate each term separately:
- \((-2\omega^2)^{100}\)
- \((-2)^{100} \times (\omega^2)^{100}\)
- \(2^{100} \times \omega^{2 \times 100}\)
- \(2^{100} \omega^{200}\)
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 \(2^{100} \omega^2\).
- \((-2\omega)^{100}\)
- \((-2)^{100} \times (\omega)^{100}\)
- \(2^{100} \omega^{100}\)
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:
\( 2^{100} \omega^2 + 2^{100} \omega \)Factor out the common term \(2^{100}\):
\( 2^{100} (\omega^2 + \omega) \)Using the property \(\omega + \omega^2 = -1\), we get:
\( 2^{100} (-1) \) \( = -2^{100} \)Final Answer
The value of the expression \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) is \(-2^{100}\).
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Important Questions from Complex Numbers
- Consider the following complex numbers
$$ z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1) $$
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