1. **State the problem:** Simplify the expression $$\left(6x y^{\frac{1}{2}}\right)^{\frac{2}{4}} \left(8x y^{\frac{3}{2}}\right)^{-4}$$. 2. **Rewrite the exponents:** Note that $$\frac{2}{4} = \frac{1}{2}$$, so the expression becomes $$\left(6x y^{\frac{1}{2}}\right)^{\frac{1}{2}} \left(8x y^{\frac{3}{2}}\right)^{-4}$$. 3. **Apply the power of a product rule:** For any terms $$a$$ and $$b$$ and exponent $$m$$, $$(ab)^m = a^m b^m$$. So, $$\left(6x y^{\frac{1}{2}}\right)^{\frac{1}{2}} = 6^{\frac{1}{2}} x^{\frac{1}{2}} \left(y^{\frac{1}{2}}\right)^{\frac{1}{2}} = 6^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{4}}$$ and $$\left(8x y^{\frac{3}{2}}\right)^{-4} = 8^{-4} x^{-4} \left(y^{\frac{3}{2}}\right)^{-4} = 8^{-4} x^{-4} y^{-6}$$. 4. **Combine the terms:** $$6^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{4}} \times 8^{-4} x^{-4} y^{-6} = 6^{\frac{1}{2}} 8^{-4} x^{\frac{1}{2} - 4} y^{\frac{1}{4} - 6}$$ 5. **Simplify the exponents:** $$x^{\frac{1}{2} - 4} = x^{-\frac{7}{2}}$$ $$y^{\frac{1}{4} - 6} = y^{-\frac{23}{4}}$$ 6. **Simplify the constants:** $$6^{\frac{1}{2}} = \sqrt{6}$$ $$8^{-4} = \frac{1}{8^4} = \frac{1}{4096}$$ 7. **Final simplified expression:** $$\frac{\sqrt{6}}{4096} x^{-\frac{7}{2}} y^{-\frac{23}{4}} = \frac{\sqrt{6}}{4096} \frac{1}{x^{\frac{7}{2}} y^{\frac{23}{4}}}$$ **Answer:** $$\boxed{\frac{\sqrt{6}}{4096 x^{\frac{7}{2}} y^{\frac{23}{4}}}}$$