1. **State the problem:** Simplify the expression $$\frac{2x^{-2} y^{-2}}{16x^{-12} y^2}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. 3. **Apply the division to coefficients:** $$\frac{2}{16} = \frac{\cancel{2}}{\cancel{16}} = \frac{1}{8}$$ 4. **Apply the division rule to $x$ terms:** $$x^{-2} \div x^{-12} = x^{-2 - (-12)} = x^{-2 + 12} = x^{10}$$ 5. **Apply the division rule to $y$ terms:** $$y^{-2} \div y^{2} = y^{-2 - 2} = y^{-4}$$ 6. **Combine all parts:** $$\frac{2x^{-2} y^{-2}}{16x^{-12} y^2} = \frac{1}{8} x^{10} y^{-4}$$ 7. **Rewrite negative exponent as positive:** $$y^{-4} = \frac{1}{y^4}$$ 8. **Final simplified expression:** $$\frac{x^{10}}{8 y^{4}}$$