1. **State the problem:** Simplify the expression $$\frac{a^5 y^3}{w y^2} \div \frac{a^3 w^2}{w^3 y^2}.$$ 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{a^5 y^3}{w y^2} \times \frac{w^3 y^2}{a^3 w^2}.$$ 3. **Combine the fractions:** $$\frac{a^5 y^3 \times w^3 y^2}{w y^2 \times a^3 w^2}.$$ 4. **Multiply the numerators and denominators:** $$\frac{a^5 w^3 y^{3+2}}{a^3 w^{1+2} y^2} = \frac{a^5 w^3 y^5}{a^3 w^3 y^2}.$$ 5. **Simplify by canceling common factors:** $$\frac{\cancel{a^3} a^{5-3} \cancel{w^3} y^{5}}{\cancel{a^3} \cancel{w^3} y^{2}} = \frac{a^{2} y^{5}}{y^{2}}.$$ 6. **Simplify the powers of y:** $$a^{2} y^{5-2} = a^{2} y^{3}.$$ **Final answer:** $$a^{2} y^{3}.$$