Question: c. $\left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\left(\frac{1}{\sqrt{2}} - \sqrt{2}\right)$
f. $\frac{2}{\sqrt{3}} \left(\frac{1}{\sqrt{3}} - \frac{\sqrt{3}}{2}\right)$
1. **Problem statement:** Simplify the expression $$\left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\left(\frac{1}{\sqrt{2}} - \sqrt{2}\right)$$.
2. **Formula used:** This is a product of conjugates, which follows the identity $$ (a+b)(a-b) = a^2 - b^2 $$.
3. **Identify terms:** Let $$a = \frac{1}{\sqrt{2}}$$ and $$b = \sqrt{2}$$.
4. **Apply the formula:**
$$\left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\left(\frac{1}{\sqrt{2}} - \sqrt{2}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 - (\sqrt{2})^2$$
5. **Calculate squares:**
$$\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{(\sqrt{2})^2} = \frac{1}{2}$$
$$ (\sqrt{2})^2 = 2 $$
6. **Substitute back:**
$$ \frac{1}{2} - 2 $$
7. **Simplify:**
$$ \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = \frac{1 - 4}{2} = \frac{-3}{2} $$
**Final answer:** $$\boxed{-\frac{3}{2}}$$