If ` a + (1)/(a) = sqrt(3)` , then the value of ` a^(6) - (1)/(a^(6)) + 2 ` will be

Options

A. 1
B. 2
C. `3 sqrt(3)`
D. 5

Answer

To solve the problem, we start with the equation given:

1. **Given**:
\[ a + \frac{1}{a} = \sqrt{3} \]

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

3. **Rearrange the equation**:
\[ a^2 + \frac{1}{a^2} = 3 - 2 = 1 \]

4. **Now, square again to find \(a^4 + \frac{1}{a^4}\)**:
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 1^2 \]
This simplifies to:
\[ a^4 + 2 + \frac{1}{a^4} = 1 \]
Rearranging gives:
\[ a^4 + \frac{1}{a^4} = 1 - 2 = -1 \]

5. **Next, we need to find \(a^6 + \frac{1}{a^6}\)**. We can use the identity:
\[ a^6 + \frac{1}{a^6} = (a^4 + \frac{1}{a^4})(a^2 + \frac{1}{a^2}) - (a^2 + \frac{1}{a^2}) \]

6. **Substituting the known values**:
\[ a^6 + \frac{1}{a^6} = (-1)(1) - 1 = -1 - 1 = -2 \]

7. **Finally, we calculate \(a^6 - \frac{1}{a^6} + 2\)**:
\[ a^6 - \frac{1}{a^6} = (a^6 + \frac{1}{a^6}) - 2 \cdot \frac{1}{a^6} \]
Since we found \(a^6 + \frac{1}{a^6} = -2\), we have:
\[ a^6 - \frac{1}{a^6} + 2 = -2 + 2 = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]