1. **State the problem:** Simplify the expression $$\frac{8 - 4\sqrt{18}}{50}$$. 2. **Simplify the square root:** Note that $$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$. 3. **Substitute back:** The numerator becomes $$8 - 4 \times 3\sqrt{2} = 8 - 12\sqrt{2}$$. 4. **Rewrite the fraction:** $$\frac{8 - 12\sqrt{2}}{50}$$. 5. **Factor numerator and denominator:** Factor out 4 from numerator: $$\frac{4(2 - 3\sqrt{2})}{50}$$. 6. **Simplify the fraction by dividing numerator and denominator by 2:** $$\frac{\cancel{4}^{2}(2 - 3\sqrt{2})}{\cancel{50}^{25}} = \frac{2(2 - 3\sqrt{2})}{25}$$. 7. **Final simplified form:** $$\frac{2(2 - 3\sqrt{2})}{25}$$ or equivalently $$\frac{4 - 6\sqrt{2}}{25}$$. This is the simplest exact form of the expression.