If `omega` is a complex cube root of unity, then what is `omega^(10) + omega^(-10)` equal to ?

Options

A. A)2
B. B)`-1`
C. C)`-2`
D. D)1

Answer

To solve the problem of finding \( \omega^{10} + \omega^{-10} \) where \( \omega \) is a complex cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The complex cube roots of unity are the solutions to the equation \( x^3 = 1 \). These roots are:
- \( 1 \)
- \( \omega \)
- \( \omega^2 \)

where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{4\pi i / 3} \). A key property of these roots is that:
\[ 1 + \omega + \omega^2 = 0 \]
This implies:
\[ \omega^2 = -1 - \omega \]

### Step 2: Simplify \( \omega^{10} \) and \( \omega^{-10} \)
Since \( \omega^3 = 1 \), we can reduce the exponent of \( \omega^{10} \) modulo 3:
\[ 10 \mod 3 = 1 \quad \Rightarrow \quad \omega^{10} = \omega^1 = \omega \]
Similarly, for \( \omega^{-10} \):
\[ -10 \mod 3 = 2 \quad \Rightarrow \quad \omega^{-10} = \omega^2 \]

### Step 3: Combine the results
Now we can substitute back into the expression:
\[ \omega^{10} + \omega^{-10} = \omega + \omega^2 \]

### Step 4: Use the property of cube roots of unity
From the property we established earlier:
\[ \omega + \omega^2 = -1 \]

### Final Result
Thus, we conclude:
\[ \omega^{10} + \omega^{-10} = -1 \]