14. \( \frac{4-\sqrt{2}}{2 \sqrt{8}} \)

To simplify \( \frac{4-\sqrt{2}}{2 \sqrt{8}} \), start by simplifying the denominator. Since \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \), our expression now looks like this: \[ \frac{4 - \sqrt{2}}{2 \cdot 2 \sqrt{2}} = \frac{4 - \sqrt{2}}{4\sqrt{2}} \] Next, we can separate it into two parts: \[ = \frac{4}{4\sqrt{2}} - \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{\sqrt{2}} - \frac{1}{4} \] Now, let's rationalize \( \frac{1}{\sqrt{2}} \): \[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] So now our expression becomes: \[ \frac{\sqrt{2}}{2} - \frac{1}{4} \] To combine these fractions, find a common denominator (which is 4): \[ \frac{2\sqrt{2}}{4} - \frac{1}{4} = \frac{2\sqrt{2} - 1}{4} \] Therefore, the simplified form of \( \frac{4-\sqrt{2}}{2 \sqrt{8}} \) is: \[ \frac{2\sqrt{2} - 1}{4} \]