If `x = 2 + sqrt(2)` and `y = 2 - sqrt(2)`, then the value of `(x^(2) - y^(2))` is

Options

A. 12
B. 14
C. `8sqrt(2)`
D. 18

Answer

To find the value of \( x^2 - y^2 \) given \( x = 2 + \sqrt{2} \) and \( y = 2 - \sqrt{2} \), we can use the difference of squares formula, which states that:

\[ a^2 - b^2 = (a + b)(a - b) \]

### Step-by-Step Solution:

1. **Identify \( x \) and \( y \)**:
\[ x = 2 + \sqrt{2}, \quad y = 2 - \sqrt{2} \]

2. **Calculate \( x + y \)**:
\[ x + y = (2 + \sqrt{2}) + (2 - \sqrt{2}) = 2 + \sqrt{2} + 2 - \sqrt{2} \]
The \( \sqrt{2} \) terms cancel out:
\[ x + y = 4 \]

3. **Calculate \( x - y \)**:
\[ x - y = (2 + \sqrt{2}) - (2 - \sqrt{2}) = 2 + \sqrt{2} - 2 + \sqrt{2} \]
The \( 2 \) terms cancel out:
\[ x - y = 2\sqrt{2} \]

4. **Apply the difference of squares formula**:
\[ x^2 - y^2 = (x + y)(x - y) = 4 \cdot (2\sqrt{2}) \]

5. **Calculate the final value**:
\[ x^2 - y^2 = 8\sqrt{2} \]

Thus, the value of \( x^2 - y^2 \) is \( 8\sqrt{2} \).