1. **State the problem:** Simplify the expression $$\frac{\sqrt{2^3 \cdot 3 \cdot a^3 \cdot b^5}}{\sqrt[3]{2^3 \cdot 3 \cdot a^3 \cdot b^5}}$$. 2. **Recall the formulas:** - Square root: $$\sqrt{x} = x^{\frac{1}{2}}$$ - Cube root: $$\sqrt[3]{x} = x^{\frac{1}{3}}$$ 3. **Rewrite the expression using exponents:** $$\frac{(2^3 \cdot 3 \cdot a^3 \cdot b^5)^{\frac{1}{2}}}{(2^3 \cdot 3 \cdot a^3 \cdot b^5)^{\frac{1}{3}}}$$ 4. **Apply the quotient rule for exponents:** $$= (2^3 \cdot 3 \cdot a^3 \cdot b^5)^{\frac{1}{2} - \frac{1}{3}}$$ 5. **Calculate the exponent difference:** $$\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$$ 6. **Simplify the expression:** $$= (2^3 \cdot 3 \cdot a^3 \cdot b^5)^{\frac{1}{6}}$$ 7. **Rewrite as a sixth root:** $$= \sqrt[6]{2^3 \cdot 3 \cdot a^3 \cdot b^5}$$ 8. **Break down inside the root:** $$= \sqrt[6]{2^3} \cdot \sqrt[6]{3} \cdot \sqrt[6]{a^3} \cdot \sqrt[6]{b^5}$$ 9. **Simplify each term:** - $$\sqrt[6]{2^3} = 2^{\frac{3}{6}} = 2^{\frac{1}{2}} = \sqrt{2}$$ - $$\sqrt[6]{3} = 3^{\frac{1}{6}}$$ (cannot simplify further) - $$\sqrt[6]{a^3} = a^{\frac{3}{6}} = a^{\frac{1}{2}} = \sqrt{a}$$ - $$\sqrt[6]{b^5} = b^{\frac{5}{6}}$$ (cannot simplify further) 10. **Combine all:** $$= \sqrt{2} \cdot 3^{\frac{1}{6}} \cdot \sqrt{a} \cdot b^{\frac{5}{6}}$$ **Final answer:** $$\boxed{\sqrt{2} \cdot 3^{\frac{1}{6}} \cdot \sqrt{a} \cdot b^{\frac{5}{6}}}$$