If `x + (1)/(x) = sqrt3` the value of `x ^(18) + x ^(-18) + 1 ` is

Options

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

Answer

To solve the problem, we need to find the value of \( x^{18} + x^{-18} + 1 \) 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 expands to:
\[ x^3 + 3\left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 3\sqrt{3} \]

3. **Substituting the value of \( x + \frac{1}{x} \):**
\[ x^3 + \frac{1}{x^3} + 3\sqrt{3} = 3\sqrt{3} \]
Simplifying gives:
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

4. **Now, we need to find \( x^6 + \frac{1}{x^6} \):**
We know that:
\[ x^3 + \frac{1}{x^3} = 0 \]
Using the identity:
\[ x^6 + \frac{1}{x^6} = \left( x^3 + \frac{1}{x^3} \right)^2 - 2 \]
Substituting:
\[ x^6 + \frac{1}{x^6} = 0^2 - 2 = -2 \]

5. **Next, we find \( x^{18} + \frac{1}{x^{18}} \):**
Using the identity again:
\[ x^{18} + \frac{1}{x^{18}} = \left( x^6 + \frac{1}{x^6} \right)^3 - 3\left( x^6 + \frac{1}{x^6} \right) \]
Substituting \( x^6 + \frac{1}{x^6} = -2 \):
\[ x^{18} + \frac{1}{x^{18}} = (-2)^3 - 3(-2) = -8 + 6 = -2 \]

6. **Finally, we compute \( x^{18} + x^{-18} + 1 \):**
\[ x^{18} + x^{-18} + 1 = -2 + 1 = -1 \]

### Final Answer:
\[ \boxed{-1} \]