If `a+ (1)/(a)= - sqrt3`, find the value of `a^(3) + (1)/(a^(3))`

Options

A. 0
B. `-6sqrt3`
C. `-3 sqrt3`
D. `6 sqrt2`

Answer

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

### Step-by-step Solution:

1. **Given Equation**:
We start with the equation:
\[ a + \frac{1}{a} = -\sqrt{3} \]

2. **Use the Formula for Cubes**:
We use the identity for cubes:
\[ a^3 + \frac{1}{a^3} = \left( a + \frac{1}{a} \right) \left( a^2 + \frac{1}{a^2} \right) - \left( a \cdot \frac{1}{a} \right) \]
Since \( a \cdot \frac{1}{a} = 1 \), we can rewrite it as:
\[ a^3 + \frac{1}{a^3} = \left( a + \frac{1}{a} \right) \left( a^2 + \frac{1}{a^2} \right) - 1 \]

3. **Find \( a^2 + \frac{1}{a^2} \)**:
To find \( a^2 + \frac{1}{a^2} \), we square \( a + \frac{1}{a} \):
\[ \left( a + \frac{1}{a} \right)^2 = a^2 + 2 + \frac{1}{a^2} \]
Therefore:
\[ a^2 + \frac{1}{a^2} = \left( a + \frac{1}{a} \right)^2 - 2 \]
Substituting \( a + \frac{1}{a} = -\sqrt{3} \):
\[ a^2 + \frac{1}{a^2} = (-\sqrt{3})^2 - 2 = 3 - 2 = 1 \]

4. **Substitute Back into the Cubes Formula**:
Now substitute \( a + \frac{1}{a} \) and \( a^2 + \frac{1}{a^2} \) back into the cubes formula:
\[ a^3 + \frac{1}{a^3} = (-\sqrt{3})(1) - 1 = -\sqrt{3} - 1 \]

5. **Final Answer**:
Thus, the value of \( a^3 + \frac{1}{a^3} \) is:
\[ a^3 + \frac{1}{a^3} = -\sqrt{3} - 1 \]

### Summary:
The value of \( a^3 + \frac{1}{a^3} \) is \( -\sqrt{3} - 1 \).