If `x + (1)/(x) = 2,` then the value of `x ^(100) + (1)/(x ^(100))` is

Options

A. 2
B. 0
C. 1
D. `-2`

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ x + \frac{1}{x} = 2 \]

2. **Rearranging the Equation**:
We can multiply both sides by \(x\) to eliminate the fraction:
\[ x^2 + 1 = 2x \]

3. **Rearranging to Form a Quadratic Equation**:
Rearranging gives us:
\[ x^2 - 2x + 1 = 0 \]

4. **Factoring the Quadratic**:
This can be factored as:
\[ (x - 1)^2 = 0 \]

5. **Finding the Value of \(x\)**:
Setting the factored equation to zero gives:
\[ x - 1 = 0 \implies x = 1 \]

6. **Finding \(x^{100} + \frac{1}{x^{100}}\)**:
Now we need to find:
\[ x^{100} + \frac{1}{x^{100}} \]
Substituting \(x = 1\):
\[ 1^{100} + \frac{1}{1^{100}} = 1 + 1 = 2 \]

7. **Final Answer**:
Therefore, the value of \(x^{100} + \frac{1}{x^{100}}\) is:
\[ \boxed{2} \]