1. **State the problem:** Simplify the expression $$\frac{3x^{-6}y^{-3}}{15x^{2}y^{10}}$$ assuming $$x \neq 0$$ and $$y \neq 0$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$. 3. **Apply the rules to the coefficients:** $$\frac{3}{15} = \frac{\cancel{3}}{\cancel{15}} = \frac{1}{5}$$ 4. **Apply the rules to the $$x$$ terms:** $$x^{-6} \div x^{2} = x^{-6-2} = x^{-8} = \frac{1}{x^{8}}$$ 5. **Apply the rules to the $$y$$ terms:** $$y^{-3} \div y^{10} = y^{-3-10} = y^{-13} = \frac{1}{y^{13}}$$ 6. **Combine all parts:** $$\frac{1}{5} \times \frac{1}{x^{8}} \times \frac{1}{y^{13}} = \frac{1}{5x^{8}y^{13}}$$ **Final answer:** $$\frac{1}{5x^{8}y^{13}}$$ which corresponds to the second option.