Question: Express as a fraction in simplest form with a rational denominator:
$\frac{-8}{7 + \sqrt{2}}$
1. **State the problem:** We need to express the fraction $$\frac{-8}{7 + \sqrt{2}}$$ in simplest form with a rational denominator.
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 $$7 + \sqrt{2}$$ is $$7 - \sqrt{2}$$.
3. **Multiply numerator and denominator by the conjugate:**
$$\frac{-8}{7 + \sqrt{2}} \times \frac{7 - \sqrt{2}}{7 - \sqrt{2}} = \frac{-8(7 - \sqrt{2})}{(7 + \sqrt{2})(7 - \sqrt{2})}$$
4. **Simplify the denominator using difference of squares:**
$$ (7 + \sqrt{2})(7 - \sqrt{2}) = 7^2 - (\sqrt{2})^2 = 49 - 2 = 47 $$
5. **Expand the numerator:**
$$ -8(7 - \sqrt{2}) = -8 \times 7 + (-8) \times (-\sqrt{2}) = -56 + 8\sqrt{2} $$
6. **Write the fraction:**
$$ \frac{-56 + 8\sqrt{2}}{47} $$
7. **Final answer:**
$$ \boxed{\frac{-56 + 8\sqrt{2}}{47}} $$
This fraction has a rational denominator and is in simplest form.