1. **State the problem:** Simplify the expression $$\frac{2^{-3} \cdot x^{0} \cdot x^{2}}{x^{-3} \cdot x \cdot y^{-3} \cdot y}$$. 2. **Recall the rules:** - Any number or variable raised to the zero power is 1, so $x^{0} = 1$. - When multiplying powers with the same base, add the exponents: $a^{m} \cdot a^{n} = a^{m+n}$. - When dividing powers with the same base, subtract the exponents: $\frac{a^{m}}{a^{n}} = a^{m-n}$. - Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^{n}}$. 3. **Simplify numerator:** $$2^{-3} \cdot x^{0} \cdot x^{2} = 2^{-3} \cdot 1 \cdot x^{2} = 2^{-3} x^{2}$$ 4. **Simplify denominator:** $$x^{-3} \cdot x \cdot y^{-3} \cdot y = x^{-3+1} \cdot y^{-3+1} = x^{-2} y^{-2}$$ 5. **Rewrite the expression:** $$\frac{2^{-3} x^{2}}{x^{-2} y^{-2}}$$ 6. **Apply division of powers:** $$2^{-3} x^{2 - (-2)} y^{0 - (-2)} = 2^{-3} x^{2+2} y^{2} = 2^{-3} x^{4} y^{2}$$ 7. **Rewrite negative exponent:** $$2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}$$ 8. **Final simplified expression:** $$\frac{x^{4} y^{2}}{8}$$ **Answer:** $$\boxed{\frac{x^{4} y^{2}}{8}}$$