1. **Simplify** $\frac{\sqrt{3} - 5\sqrt{2}}{5\sqrt{3}}$. Use the property $\frac{a - b}{c} = \frac{a}{c} - \frac{b}{c}$: $$\frac{\sqrt{3}}{5\sqrt{3}} - \frac{5\sqrt{2}}{5\sqrt{3}}$$ Simplify each term: $$\frac{\cancel{\sqrt{3}}}{5\cancel{\sqrt{3}}} - \frac{\cancel{5}\sqrt{2}}{\cancel{5}\sqrt{3}} = \frac{1}{5} - \frac{\sqrt{2}}{\sqrt{3}}$$ Rationalize the second term: $$\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3}$$ So the expression becomes: $$\frac{1}{5} - \frac{\sqrt{6}}{3}$$ Find common denominator $15$: $$\frac{3}{15} - \frac{5\sqrt{6}}{15} = \frac{3 - 5\sqrt{6}}{15}$$ **Final answer:** $$\boxed{\frac{3 - 5\sqrt{6}}{15}}$$