If `x-(2)/(sqrt(10)-sqrt(8)),y-(2)/(sqrt(10)+2sqrt(2))`, then `(x-y)^(2)`=

Options

A. `4sqrt(2)`
B. 32
C. `8sqrt(2)`
D. 64

Explanation

To solve the problem, we need to find \((x - y)^2\) given the values of \(x\) and \(y\):

1. **Define \(x\) and \(y\)**:
\[ x = \frac{2}{\sqrt{10} - \sqrt{8}}, \quad y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \]

2. **Rationalize \(x\)**:
To rationalize \(x\), multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{2}{\sqrt{10} - \sqrt{8}} \cdot \frac{\sqrt{10} + \sqrt{8}}{\sqrt{10} + \sqrt{8}} = \frac{2(\sqrt{10} + \sqrt{8})}{(\sqrt{10})^2 - (\sqrt{8})^2} \]
Calculate the denominator:
\[ (\sqrt{10})^2 - (\sqrt{8})^2 = 10 - 8 = 2 \]
Thus,
\[ x = \frac{2(\sqrt{10} + \sqrt{8})}{2} = \sqrt{10} + \sqrt{8} \]

3. **Rationalize \(y\)**:
Similarly, rationalize \(y\):
\[ y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \cdot \frac{\sqrt{10} - 2\sqrt{2}}{\sqrt{10} - 2\sqrt{2}} = \frac{2(\sqrt{10} - 2\sqrt{2})}{(\sqrt{10})^2 - (2\sqrt{2})^2} \]
Calculate the denominator:
\[ (\sqrt{10})^2 - (2\sqrt{2})^2 = 10 - 8 = 2 \]
Thus,
\[ y = \frac{2(\sqrt{10} - 2\sqrt{2})}{2} = \sqrt{10} - 2\sqrt{2} \]

4. **Calculate \(x - y\)**:
Now, find \(x - y\):
\[ x - y = (\sqrt{10} + \sqrt{8}) - (\sqrt{10} - 2\sqrt{2}) = \sqrt{8} + 2\sqrt{2} \]
Since \(\sqrt{8} = 2\sqrt{2}\), we have:
\[ x - y = 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2} \]

5. **Calculate \((x - y)^2\)**:
Now, we need to square \(x - y\):
\[ (x - y)^2 = (4\sqrt{2})^2 = 16 \cdot 2 = 32 \]

Thus, the final answer is:
\[ \boxed{32} \]