If `x+ (1)/(x)=2`, find
`x^(11) + (1)/(x^(11))=` ?

Answer

To solve the equation \( x + \frac{1}{x} = 2 \) and find \( x^{11} + \frac{1}{x^{11}} \), we can follow these steps:

### Step 1: Solve for \( x \)
Given the equation:
\[ x + \frac{1}{x} = 2 \]

We can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2x \]

Rearranging gives us:
\[ x^2 - 2x + 1 = 0 \]

### Step 2: Factor the quadratic
The quadratic can be factored as:
\[ (x - 1)^2 = 0 \]

### Step 3: Solve for \( x \)
Setting the factor equal to zero gives:
\[ x - 1 = 0 \implies x = 1 \]

### Step 4: Find \( \frac{1}{x} \)
Since \( x = 1 \):
\[ \frac{1}{x} = \frac{1}{1} = 1 \]

### Step 5: Calculate \( x^{11} + \frac{1}{x^{11}} \)
Now we can substitute \( x \) into the expression we need to find:
\[ x^{11} + \frac{1}{x^{11}} = 1^{11} + \frac{1}{1^{11}} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( x^{11} + \frac{1}{x^{11}} \) is:
\[ \boxed{2} \]