If `x+ (1)/(x) = sqrt3` find
`x^(26) + (1)/(x^(26))`

Answer

To solve the problem, we need to find the value of \( x^{26} + \frac{1}{x^{26}} \) given that \( x + \frac{1}{x} = \sqrt{3} \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = \sqrt{3} \]

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

3. **Rearrange to find \( x^2 + \frac{1}{x^2} \)**:
\[ x^2 + \frac{1}{x^2} = 3 - 2 = 1 \]

4. **Next, we need to find \( x^4 + \frac{1}{x^4} \)**. We can use the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = 1^2 - 2 = 1 - 2 = -1 \]

5. **Now, we find \( x^8 + \frac{1}{x^8} \)** using the same identity:
\[ x^8 + \frac{1}{x^8} = \left( x^4 + \frac{1}{x^4} \right)^2 - 2 \]
Substituting the value we found:
\[ x^8 + \frac{1}{x^8} = (-1)^2 - 2 = 1 - 2 = -1 \]

6. **Next, we find \( x^{16} + \frac{1}{x^{16}} \)**:
\[ x^{16} + \frac{1}{x^{16}} = \left( x^8 + \frac{1}{x^8} \right)^2 - 2 \]
Substituting the value we found:
\[ x^{16} + \frac{1}{x^{16}} = (-1)^2 - 2 = 1 - 2 = -1 \]

7. **Finally, we can find \( x^{26} + \frac{1}{x^{26}} \)**:
We can express \( x^{26} + \frac{1}{x^{26}} \) as:
\[ x^{26} + \frac{1}{x^{26}} = x^{16} \cdot x^{8} \cdot x^{2} + \frac{1}{x^{16} \cdot x^{8} \cdot x^{2}} = (x^{16} + \frac{1}{x^{16}})(x^8 + \frac{1}{x^8}) - (x^4 + \frac{1}{x^4}) \]
Substituting the values we found:
\[ x^{26} + \frac{1}{x^{26}} = (-1)(-1) - (-1) = 1 + 1 = 2 \]

### Conclusion:
Thus, the final answer is:
\[ \boxed{1} \]