1. **State the problem:** Simplify the expression $$\frac{8 - 4 \sqrt{18}}{50}$$. 2. **Recall the rules:** Simplify the square root first, then simplify the fraction by factoring and canceling common factors. 3. **Simplify the square root:** $$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}$$. 4. **Substitute back:** $$8 - 4 \sqrt{18} = 8 - 4 \times 3 \sqrt{2} = 8 - 12 \sqrt{2}$$. 5. **Rewrite the fraction:** $$\frac{8 - 12 \sqrt{2}}{50}$$. 6. **Factor numerator and denominator:** Numerator: $$4(2 - 3 \sqrt{2})$$, Denominator: $$50 = 2 \times 25$$. 7. **Simplify the fraction:** $$\frac{\cancel{4}(2 - 3 \sqrt{2})}{\cancel{50}} = \frac{2 - 3 \sqrt{2}}{12.5}$$ (since $$\frac{4}{50} = \frac{2}{25}$$, better to write as $$\frac{4(2 - 3 \sqrt{2})}{50} = \frac{2 - 3 \sqrt{2}}{12.5}$$). 8. **Express denominator as fraction:** $$12.5 = \frac{25}{2}$$, so $$\frac{2 - 3 \sqrt{2}}{12.5} = (2 - 3 \sqrt{2}) \times \frac{2}{25} = \frac{2(2 - 3 \sqrt{2})}{25}$$. 9. **Final simplified form:** $$\boxed{\frac{4 - 6 \sqrt{2}}{25}}$$. This is the simplest exact form of the expression.