If x = 3 + `2 sqrt(2)`, find the value fo `sqrt(2) (x^(2) - x^(-2))`.

Options

A. 24
B. `24 sqrt(2)`
C. 48
D. `96 sqrt(2)`

Answer

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

### Step-by-Step Solution:

1. **Find \( x^{-1} \)**:
\[ x^{-1} = \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \]
To rationalize the denominator, multiply the numerator and denominator by the conjugate \( 3 - 2\sqrt{2} \):
\[ x^{-1} = \frac{3 - 2\sqrt{2}}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]
The denominator simplifies as follows:
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
Thus, we have:
\[ x^{-1} = 3 - 2\sqrt{2} \]

2. **Calculate \( x + x^{-1} \)**:
\[ x + x^{-1} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]

3. **Calculate \( x - x^{-1} \)**:
\[ x - x^{-1} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) = 4\sqrt{2} \]

4. **Use the identity for \( x^2 - x^{-2} \)**:
We know that:
\[ x^2 - x^{-2} = (x - x^{-1})(x + x^{-1}) \]
Substituting the values we found:
\[ x^2 - x^{-2} = (4\sqrt{2})(6) = 24\sqrt{2} \]

5. **Calculate \( \sqrt{2}(x^2 - x^{-2}) \)**:
\[ \sqrt{2}(x^2 - x^{-2}) = \sqrt{2}(24\sqrt{2}) = 24 \cdot 2 = 48 \]

### Final Answer:
The value of \( \sqrt{2}(x^2 - x^{-2}) \) is \( \boxed{48} \).