If `(x+(1)/(x))=2sqrt3`, then the value of `(x^(3)-(1)/(x^(3)))` is :

Options

A. `12sqrt3`
B. 18
C. `18sqrt3`
D. none of these

Answer

To solve the problem, we need to find the value of \( x^3 - \frac{1}{x^3} \) given that \( x + \frac{1}{x} = 2\sqrt{3} \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + \frac{1}{x} = 2\sqrt{3} \]

2. **Square both sides**:
\[ \left( x + \frac{1}{x} \right)^2 = (2\sqrt{3})^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 12 \]

3. **Rearrange the equation**:
\[ x^2 + \frac{1}{x^2} = 12 - 2 = 10 \]

4. **Use the identity for \( x^3 - \frac{1}{x^3} \)**:
The identity states:
\[ x^3 - \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 - 1 + \frac{1}{x^2} \right) \]
We already have \( x + \frac{1}{x} = 2\sqrt{3} \) and we need \( x^2 - 1 + \frac{1}{x^2} \).

5. **Calculate \( x^2 - 1 + \frac{1}{x^2} \)**:
From step 3, we know:
\[ x^2 + \frac{1}{x^2} = 10 \]
Thus:
\[ x^2 - 1 + \frac{1}{x^2} = 10 - 1 = 9 \]

6. **Substitute back into the identity**:
Now substitute back into the identity:
\[ x^3 - \frac{1}{x^3} = (2\sqrt{3})(9) \]
This simplifies to:
\[ x^3 - \frac{1}{x^3} = 18\sqrt{3} \]

### Final Answer:
\[ x^3 - \frac{1}{x^3} = 18\sqrt{3} \]