If ` x + (1)/( x) = sqrt(3)` then the value of ` x^(18) + x ^(12) + x^(6) + 1 ` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( x + \frac{1}{x} = \sqrt{3} \) and find the value of \( x^{18} + x^{12} + x^{6} + 1 \), we can follow these steps:

### Step 1: Cube the equation
We start with the equation:
\[ x + \frac{1}{x} = \sqrt{3} \]
Now, we will cube both sides:
\[ \left( x + \frac{1}{x} \right)^3 = (\sqrt{3})^3 \]
This expands to:
\[ x^3 + 3x \cdot \frac{1}{x} \left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 3\sqrt{3} \]
Simplifying the middle term:
\[ x^3 + 3\left( x + \frac{1}{x} \right) + \frac{1}{x^3} = 3\sqrt{3} \]
Substituting \( x + \frac{1}{x} = \sqrt{3} \):
\[ x^3 + 3\sqrt{3} + \frac{1}{x^3} = 3\sqrt{3} \]
Thus, we have:
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

### Step 2: Solve for \( x^3 \)
From \( x^3 + \frac{1}{x^3} = 0 \), we can rearrange this to:
\[ x^3 = -\frac{1}{x^3} \]
Multiplying both sides by \( x^3 \):
\[ x^6 = -1 \]

### Step 3: Substitute into the expression
Now we need to find the value of:
\[ x^{18} + x^{12} + x^{6} + 1 \]
We can express these powers in terms of \( x^6 \):
\[ x^{18} = (x^6)^3 = (-1)^3 = -1 \]
\[ x^{12} = (x^6)^2 = (-1)^2 = 1 \]
\[ x^{6} = -1 \]
Now substituting these values into the expression:
\[ x^{18} + x^{12} + x^{6} + 1 = -1 + 1 - 1 + 1 \]

### Step 4: Simplify the expression
Calculating the above:
\[ -1 + 1 - 1 + 1 = 0 \]

### Final Answer
Thus, the value of \( x^{18} + x^{12} + x^{6} + 1 \) is:
\[ \boxed{0} \]