What is `omega^(100) + omega^(200) + omega^(300)` equal to, where `omega` is the cube root of unity ?

Options

A. 1
B. `3 omega`
C. `3 omega^(2)`
D. 0

Answer

To solve the problem of finding the value of \( \omega^{100} + \omega^{200} + \omega^{300} \), where \( \omega \) is a cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). They are:
- \( 1 \)
- \( \omega = e^{2\pi i / 3} = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \)
- \( \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \)

These roots satisfy the following properties:
1. \( 1 + \omega + \omega^2 = 0 \)
2. \( \omega^3 = 1 \)

### Step 2: Reduce the powers of \( \omega \)
Since \( \omega^3 = 1 \), we can reduce the exponents of \( \omega \) modulo 3:
- \( \omega^{100} = \omega^{100 \mod 3} = \omega^{1} = \omega \)
- \( \omega^{200} = \omega^{200 \mod 3} = \omega^{2} \)
- \( \omega^{300} = \omega^{300 \mod 3} = \omega^{0} = 1 \)

### Step 3: Substitute the reduced values into the expression
Now we can substitute these values back into the original expression:
\[ \omega^{100} + \omega^{200} + \omega^{300} = \omega + \omega^2 + 1 \]

### Step 4: Use the property of cube roots of unity
From the property \( 1 + \omega + \omega^2 = 0 \), we can conclude:
\[ \omega + \omega^2 + 1 = 0 \]

### Final Answer
Thus, the value of \( \omega^{100} + \omega^{200} + \omega^{300} \) is:
\[ \boxed{0} \]