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

Options

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

Explanation

To solve the expression \( \omega^{100} + \omega^{200} + \omega^{300} \), 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 \( x^3 = 1 \). The three roots are:
- \( 1 \)
- \( \omega \)
- \( \omega^2 \)

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

### Step 2: Reduce the exponents modulo 3
Since \( \omega^3 = 1 \), we can reduce the exponents of \( \omega \) modulo 3:
- \( 100 \mod 3 = 1 \) (because \( 100 = 3 \times 33 + 1 \))
- \( 200 \mod 3 = 2 \) (because \( 200 = 3 \times 66 + 2 \))
- \( 300 \mod 3 = 0 \) (because \( 300 = 3 \times 100 + 0 \))

Thus, we can rewrite the powers:
- \( \omega^{100} = \omega^1 = \omega \)
- \( \omega^{200} = \omega^2 \)
- \( \omega^{300} = \omega^0 = 1 \)

### Step 3: Substitute back into the expression
Now we can substitute these results 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 we mentioned earlier, we know:
\[ 1 + \omega + \omega^2 = 0 \]

### Step 5: Conclude the result
Thus, we have:
\[ \omega + \omega^2 + 1 = 0 \]

Therefore, the final answer is:
\[ \omega^{100} + \omega^{200} + \omega^{300} = 0 \]

---