Certainly! Here is the numerical solution:

1. Solve for \( x \):
   \[
   \sqrt[6]{2\sqrt{2}} = x
   \]
   \[
   x = 2^{1/4}
   \]
   \[
   x = \sqrt[4]{2}
   \]

2. Evaluate the expression \( \frac{2 - x^2}{2 + x^2} \):
   \[
   x^2 = (\sqrt[4]{2})^2 = \sqrt{2}
   \]
   \[
   \frac{2 - \sqrt{2}}{2 + \sqrt{2}}
   \]

3. Simplify the fraction by multiplying by the conjugate of the denominator:
   \[
   \frac{2 - \sqrt{2}}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{(2 - \sqrt{2})^2}{(2 + \sqrt{2})(2 - \sqrt{2})}
   \]
   \[
   \frac{6 - 4\sqrt{2}}{2} = 3 - 2\sqrt{2}
   \]

The answer is:

**A) \( 3 - 2\sqrt{2} \)**

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