1. **State the problem:** Simplify the expression $$\left( \frac{16x^{5}y^{10}}{81xy^{2}} \right)^{\frac{3}{4}}$$ assuming all variables are positive. 2. **Simplify inside the parentheses first:** $$\frac{16x^{5}y^{10}}{81xy^{2}} = \frac{16}{81} \cdot \frac{x^{5}}{x} \cdot \frac{y^{10}}{y^{2}} = \frac{16}{81} x^{5-1} y^{10-2} = \frac{16}{81} x^{4} y^{8}$$ 3. **Rewrite the expression:** $$\left( \frac{16}{81} x^{4} y^{8} \right)^{\frac{3}{4}}$$ 4. **Apply the exponent to each factor:** $$\left( \frac{16}{81} \right)^{\frac{3}{4}} \cdot \left( x^{4} \right)^{\frac{3}{4}} \cdot \left( y^{8} \right)^{\frac{3}{4}}$$ 5. **Simplify each term:** - For the fraction: $$\left( \frac{16}{81} \right)^{\frac{3}{4}} = \frac{16^{\frac{3}{4}}}{81^{\frac{3}{4}}}$$ - Calculate numerator: $$16^{\frac{3}{4}} = \left(16^{\frac{1}{4}}\right)^{3} = (2)^{3} = 8$$ - Calculate denominator: $$81^{\frac{3}{4}} = \left(81^{\frac{1}{4}}\right)^{3} = (3)^{3} = 27$$ - For the variables: $$\left( x^{4} \right)^{\frac{3}{4}} = x^{4 \cdot \frac{3}{4}} = x^{3}$$ $$\left( y^{8} \right)^{\frac{3}{4}} = y^{8 \cdot \frac{3}{4}} = y^{6}$$ 6. **Combine all simplified parts:** $$\frac{8}{27} x^{3} y^{6}$$ **Final answer:** $$\boxed{\frac{8}{27} x^{3} y^{6}}$$