Question: Express as a fraction in simplest form with a rational denominator: $$\frac{10}{5 - \sqrt{7}}$$
1. **State the problem:** Simplify the expression $$\frac{10}{5 - \sqrt{7}}$$ so that the denominator is rational.
2. **Formula and rule:** To rationalize the denominator when it contains a square root, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$5 - \sqrt{7}$$ is $$5 + \sqrt{7}$$.
3. **Multiply numerator and denominator by the conjugate:**
$$\frac{10}{5 - \sqrt{7}} \times \frac{5 + \sqrt{7}}{5 + \sqrt{7}} = \frac{10(5 + \sqrt{7})}{(5 - \sqrt{7})(5 + \sqrt{7})}$$
4. **Simplify the denominator using difference of squares:**
$$ (5 - \sqrt{7})(5 + \sqrt{7}) = 5^2 - (\sqrt{7})^2 = 25 - 7 = 18 $$
5. **Write the numerator:**
$$ 10(5 + \sqrt{7}) = 50 + 10\sqrt{7} $$
6. **Put it all together:**
$$ \frac{50 + 10\sqrt{7}}{18} $$
7. **Simplify the fraction by dividing numerator and denominator by 2:**
$$ \frac{\cancel{2} \times (25 + 5\sqrt{7})}{\cancel{2} \times 9} = \frac{25 + 5\sqrt{7}}{9} $$
8. **Final answer:**
$$ \boxed{\frac{25 + 5\sqrt{7}}{9}} $$