If `x = 2 -sqrt(3)` then the value of `x^(3)-x^(-3)` is :

Options

A. `-30 sqrt(3)`
B. `30 sqrt(3)`
C. `-30 sqrt(2)`
D. `30 sqrt(2)`

Answer

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

### Step-by-Step Solution:

1. **Find \( x^{-1} \)**:
\[ x^{-1} = \frac{1}{x} = \frac{1}{2 - \sqrt{3}} \]
To rationalize the denominator, multiply the numerator and denominator by the conjugate:
\[ x^{-1} = \frac{1 \cdot (2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3} \]

2. **Calculate \( x - x^{-1} \)**:
\[ x - x^{-1} = (2 - \sqrt{3}) - (2 + \sqrt{3}) = 2 - \sqrt{3} - 2 - \sqrt{3} = -2\sqrt{3} \]

3. **Let \( a = x - x^{-1} \)**:
\[ a = -2\sqrt{3} \]

4. **Use the identity for \( x^3 - x^{-3} \)**:
The formula for \( x^3 - x^{-3} \) in terms of \( a \) is:
\[ x^3 - x^{-3} = (x - x^{-1})^3 + 3(x - x^{-1}) \]
Substituting \( a \):
\[ x^3 - x^{-3} = a^3 + 3a \]

5. **Calculate \( a^3 \)**:
\[ a^3 = (-2\sqrt{3})^3 = -8 \cdot 3\sqrt{3} = -24\sqrt{3} \]

6. **Calculate \( 3a \)**:
\[ 3a = 3 \cdot (-2\sqrt{3}) = -6\sqrt{3} \]

7. **Combine the results**:
\[ x^3 - x^{-3} = -24\sqrt{3} - 6\sqrt{3} = -30\sqrt{3} \]

### Final Answer:
\[ x^3 - x^{-3} = -30\sqrt{3} \]