If `x+((1)/(x))=2`, then what is the value of `x^(21)+((1)/(x^(1331)))`?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the equation \( x + \frac{1}{x} = 2 \) and find the value of \( x^{21} + \frac{1}{x^{1331}} \), 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:
\[ x^2 - 2x + 1 = 0 \]
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

### Step 2: Substitute \( x \) into the expression
Now we need to evaluate:
\[ x^{21} + \frac{1}{x^{1331}} \]
Substituting \( x = 1 \):
\[ 1^{21} + \frac{1}{1^{1331}} = 1 + 1 = 2 \]

### Conclusion
The value of \( x^{21} + \frac{1}{x^{1331}} \) is:
\[ \boxed{2} \]