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

Options

A. `22sqrt(3)`
B. `-22sqrt(2)`
C. `22sqrt(2)`
D. `-22sqrt(3)`

Answer

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

### Step 1: Rewrite the expression
We can rewrite \( x^3 - x^{-3} \) as:
\[ x^3 - x^{-3} = x^3 - \frac{1}{x^3} \]
This can be expressed using the identity:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x}) \left( (x^2 + 1 + \frac{1}{x^2}) \right) \]

### Step 2: Calculate \( x - \frac{1}{x} \)
First, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\sqrt{3} - \sqrt{2}} \]
To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{3} + \sqrt{2} \):
\[ \frac{1}{x} = \frac{\sqrt{3} + \sqrt{2}}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} = \frac{\sqrt{3} + \sqrt{2}}{3 - 2} = \sqrt{3} + \sqrt{2} \]
Now, we can calculate \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = (\sqrt{3} - \sqrt{2}) - (\sqrt{3} + \sqrt{2}) = -2\sqrt{2} \]

### Step 3: Calculate \( x^2 + \frac{1}{x^2} \)
Next, we need to find \( x^2 + \frac{1}{x^2} \). We can use the identity:
\[ x^2 + \frac{1}{x^2} = (x - \frac{1}{x})^2 + 2 \]
Substituting \( x - \frac{1}{x} = -2\sqrt{2} \):
\[ x^2 + \frac{1}{x^2} = (-2\sqrt{2})^2 + 2 = 8 + 2 = 10 \]

### Step 4: Calculate \( x^3 - x^{-3} \)
Now we can substitute back into our expression for \( x^3 - x^{-3} \):
\[ x^3 - x^{-3} = (x - \frac{1}{x}) \left( x^2 + 1 + \frac{1}{x^2} \right) \]
Substituting \( x - \frac{1}{x} = -2\sqrt{2} \) and \( x^2 + \frac{1}{x^2} = 10 \):
\[ x^3 - x^{-3} = (-2\sqrt{2}) \left( 10 + 1 \right) = (-2\sqrt{2}) \cdot 11 = -22\sqrt{2} \]

### Final Answer
Thus, the value of \( x^3 - x^{-3} \) is:
\[ \boxed{-22\sqrt{2}} \]