1. **State the problem:** Simplify the expression $$\left(\frac{x^{5} y^{-3}}{2x y^{5}}\right)^{-4} \div \frac{4x^{6} y^{-10}}{\left(3x^{-2} y^{2}\right)^{-3}}.$$\n\n2. **Rewrite the expression clearly:**\n$$\left(\frac{x^{5} y^{-3}}{2x y^{5}}\right)^{-4} \div \frac{4x^{6} y^{-10}}{\left(3x^{-2} y^{2}\right)^{-3}} = \left(\frac{x^{5} y^{-3}}{2x y^{5}}\right)^{-4} \times \frac{\left(3x^{-2} y^{2}\right)^{3}}{4x^{6} y^{-10}}.$$\n\n3. **Simplify inside the first parentheses:**\n$$\frac{x^{5} y^{-3}}{2x y^{5}} = \frac{x^{5}}{x} \times \frac{y^{-3}}{y^{5}} \times \frac{1}{2} = \frac{x^{5-1}}{2} \times y^{-3-5} = \frac{x^{4}}{2} y^{-8}.$$\n\n4. **Apply the exponent -4:**\n$$\left(\frac{x^{4}}{2} y^{-8}\right)^{-4} = \left(\frac{x^{4} y^{-8}}{2}\right)^{-4} = \left(\frac{x^{4} y^{-8}}{2}\right)^{-4} = \left(\frac{2}{x^{4} y^{-8}}\right)^{4} = \frac{2^{4}}{x^{16} y^{-32}} = \frac{16}{x^{16} y^{-32}}.$$\n\n5. **Simplify the second big fraction denominator:**\n$$\left(3x^{-2} y^{2}\right)^{-3} = 3^{-3} x^{6} y^{-6} = \frac{1}{27} x^{6} y^{-6}.$$\n\n6. **Rewrite the second fraction:**\n$$\frac{4x^{6} y^{-10}}{\left(3x^{-2} y^{2}\right)^{-3}} = \frac{4x^{6} y^{-10}}{\frac{1}{27} x^{6} y^{-6}} = 4x^{6} y^{-10} \times \frac{27}{x^{6} y^{-6}} = 4 \times 27 \times \frac{x^{6}}{x^{6}} \times \frac{y^{-10}}{y^{-6}} = 108 y^{-4}.$$\n\n7. **Now the entire expression is:**\n$$\frac{16}{x^{16} y^{-32}} \div 108 y^{-4} = \frac{16}{x^{16} y^{-32}} \times \frac{1}{108 y^{-4}} = \frac{16}{108} \times \frac{1}{x^{16}} \times \frac{1}{y^{-32} y^{-4}} = \frac{16}{108} \times x^{-16} \times y^{32+4} = \frac{16}{108} x^{-16} y^{36}.$$\n\n8. **Simplify the fraction:**\n$$\frac{16}{108} = \frac{\cancel{16}^{4}}{\cancel{108}^{27}} = \frac{4}{27}.$$\n\n9. **Final simplified expression:**\n$$\boxed{\frac{4}{27} x^{-16} y^{36}}.$$\n\nThis means the expression simplifies to $$\frac{4}{27} x^{-16} y^{36}$$ which can also be written as $$\frac{4 y^{36}}{27 x^{16}}.$$