If `x+ (1)/(x) = sqrt3` find
`x^(102) + (1)/(x^(102))`

Answer

To solve the problem, we need to find \( x^{102} + \frac{1}{x^{102}} \) 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. **Cube both sides:**
\[ \left( x + \frac{1}{x} \right)^3 = (\sqrt{3})^3 \]
This simplifies to:
\[ x^3 + 3x \cdot \frac{1}{x}(x + \frac{1}{x}) + \frac{1}{x^3} = 3\sqrt{3} \]
Which can be rewritten as:
\[ x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) = 3\sqrt{3} \]

3. **Substitute \( x + \frac{1}{x} \):**
Substitute \( \sqrt{3} \) into the equation:
\[ x^3 + \frac{1}{x^3} + 3\sqrt{3} = 3\sqrt{3} \]

4. **Simplify the equation:**
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

5. **Now we know:**
\[ x^3 + \frac{1}{x^3} = 0 \]

6. **Next, we need to find \( x^{102} + \frac{1}{x^{102}} \):**
We can express \( x^{102} \) in terms of \( x^3 \):
\[ x^{102} = (x^3)^{34} \quad \text{and} \quad \frac{1}{x^{102}} = \left( \frac{1}{x^3} \right)^{34} \]

7. **Using the result from step 4:**
Since \( x^3 + \frac{1}{x^3} = 0 \), we can find:
\[ x^{102} + \frac{1}{x^{102}} = (x^3)^{34} + \left( \frac{1}{x^3} \right)^{34} \]

8. **Evaluate \( (x^3)^{34} + \left( \frac{1}{x^3} \right)^{34} \):**
Since \( x^3 + \frac{1}{x^3} = 0 \), we can deduce:
\[ (x^3)^{34} + \left( \frac{1}{x^3} \right)^{34} = 0^{34} = 0 \]

9. **Final result:**
Therefore, we conclude:
\[ x^{102} + \frac{1}{x^{102}} = -2 \]

### Final Answer:
\[ x^{102} + \frac{1}{x^{102}} = -2 \]