If `(x+1/x)`=2 then the value of `(x^5+1/x^5)` is:

Options

A. 2
B. 4
C. 8
D. none of these

Answer

To solve the problem, we need to find the value of \( x^5 + \frac{1}{x^5} \) given that \( x + \frac{1}{x} = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 2 \]

2. **Rearranging the equation:**
We can rewrite the equation to isolate \( x \):
\[ x + \frac{1}{x} - 2 = 0 \]

3. **Finding the value of \( x \):**
To find \( x \), we can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 - 2x + 1 = 0 \]
This simplifies to:
\[ (x - 1)^2 = 0 \]
Therefore, we find:
\[ x = 1 \]

4. **Calculating \( x^5 + \frac{1}{x^5} \):**
Now that we have \( x = 1 \), we can substitute this value into \( x^5 + \frac{1}{x^5} \):
\[ x^5 + \frac{1}{x^5} = 1^5 + \frac{1}{1^5} = 1 + 1 = 2 \]

5. **Final answer:**
Thus, the value of \( x^5 + \frac{1}{x^5} \) is:
\[ \boxed{2} \]