1. **State the problem:** Simplify the expression $$\left( \frac{4x^{3} y^{-4}}{16x^{-3} y^{4}} \right)^{-\frac{1}{2}}$$. 2. **Rewrite the fraction inside the parentheses:** $$\frac{4x^{3} y^{-4}}{16x^{-3} y^{4}} = \frac{4}{16} \cdot \frac{x^{3}}{x^{-3}} \cdot \frac{y^{-4}}{y^{4}}$$ 3. **Simplify each part:** - Simplify the coefficients: $$\frac{4}{16} = \frac{\cancel{4}}{4 \times \cancel{4}} = \frac{1}{4}$$ - Simplify the $x$ terms using the rule $$\frac{x^{a}}{x^{b}} = x^{a-b}$$: $$x^{3 - (-3)} = x^{3 + 3} = x^{6}$$ - Simplify the $y$ terms similarly: $$y^{-4 - 4} = y^{-8}$$ 4. **Combine the simplified parts:** $$\frac{1}{4} x^{6} y^{-8}$$ 5. **Rewrite the original expression with the simplified base:** $$\left( \frac{1}{4} x^{6} y^{-8} \right)^{-\frac{1}{2}}$$ 6. **Apply the power of a product rule:** $$\left(a b c\right)^{m} = a^{m} b^{m} c^{m}$$ 7. **Apply the exponent to each factor:** $$\left( \frac{1}{4} \right)^{-\frac{1}{2}} x^{6 \times -\frac{1}{2}} y^{-8 \times -\frac{1}{2}}$$ 8. **Simplify each term:** - $$\left( \frac{1}{4} \right)^{-\frac{1}{2}} = \left(4\right)^{\frac{1}{2}} = 2$$ - $$x^{6 \times -\frac{1}{2}} = x^{-3} = \frac{1}{x^{3}}$$ - $$y^{-8 \times -\frac{1}{2}} = y^{4}$$ 9. **Combine all terms:** $$2 \cdot \frac{1}{x^{3}} \cdot y^{4} = \frac{2 y^{4}}{x^{3}}$$ **Final answer:** $$\boxed{\frac{2 y^{4}}{x^{3}}}$$