If `z + z^(-1)= 1`, then find the value of `z^(100) + z^(-100)`.

Options

A. i
B. `-i`
C. 1
D. `-1`

Answer

To solve the equation \( z + z^{-1} = 1 \) and find the value of \( z^{100} + z^{-100} \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ z + z^{-1} = 1 \]
Multiply both sides by \( z \) to eliminate the fraction:
\[ z^2 + 1 = z \]

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ z^2 - z + 1 = 0 \]

### Step 3: Use the quadratic formula
We can solve for \( z \) using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = 1 \). Plugging in these values:
\[ z = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ z = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 4: Identify the roots
The roots can be expressed as:
\[ z_1 = \frac{1 + i\sqrt{3}}{2}, \quad z_2 = \frac{1 - i\sqrt{3}}{2} \]
These roots correspond to the complex numbers \( \omega \) and \( \omega^2 \), where \( \omega = e^{i\pi/3} \) and \( \omega^2 = e^{-i\pi/3} \).

### Step 5: Find \( z^{100} + z^{-100} \)
We can express \( z^{100} + z^{-100} \) using the properties of roots of unity. Since \( z_1 \) and \( z_2 \) are the cube roots of unity, we have:
\[ z^{100} = (z^3)^{33} \cdot z = 1^{33} \cdot z = z \]
\[ z^{-100} = (z^{-3})^{33} \cdot z^{-1} = 1^{33} \cdot z^{-1} = z^{-1} \]
Thus:
\[ z^{100} + z^{-100} = z + z^{-1} \]
From our original equation, we know:
\[ z + z^{-1} = 1 \]

### Final Answer
Therefore, the value of \( z^{100} + z^{-100} \) is:
\[ \boxed{1} \]