If `(x+(1)/(x))=2`, then the value of `(x^(6)+(1)/(x^(6)))` is :

Options

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

Answer

To solve the problem, we need to find the value of \( x^6 + \frac{1}{x^6} \) given that \( x + \frac{1}{x} = 2 \).

### Step-by-step Solution:

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

2. **Cube both sides:**
\[ \left( x + \frac{1}{x} \right)^3 = 2^3 \]
This gives us:
\[ x^3 + \frac{1}{x^3} + 3\left( x + \frac{1}{x} \right) = 8 \]

3. **Substitute the value of \( x + \frac{1}{x} \):**
\[ x^3 + \frac{1}{x^3} + 3 \cdot 2 = 8 \]
Simplifying this:
\[ x^3 + \frac{1}{x^3} + 6 = 8 \]

4. **Isolate \( x^3 + \frac{1}{x^3} \):**
\[ x^3 + \frac{1}{x^3} = 8 - 6 = 2 \]

5. **Square both sides to find \( x^6 + \frac{1}{x^6} \):**
\[ \left( x^3 + \frac{1}{x^3} \right)^2 = 2^2 \]
This expands to:
\[ x^6 + \frac{1}{x^6} + 2 = 4 \]

6. **Isolate \( x^6 + \frac{1}{x^6} \):**
\[ x^6 + \frac{1}{x^6} = 4 - 2 = 2 \]

### Final Answer:
\[ x^6 + \frac{1}{x^6} = 2 \]