1. **State the problem:** Simplify the radical expression $$\sqrt[3]{\frac{81x^{2}y^{8}}{3x^{8}y^{2}}}$$ and find which option is equivalent. 2. **Write the expression inside the cube root as a single fraction:** $$\frac{81x^{2}y^{8}}{3x^{8}y^{2}}$$ 3. **Simplify the fraction by dividing coefficients and subtracting exponents of like bases:** - Coefficients: $$\frac{81}{3} = 27$$ - For $$x$$: $$x^{2} / x^{8} = x^{2-8} = x^{-6}$$ - For $$y$$: $$y^{8} / y^{2} = y^{8-2} = y^{6}$$ So the expression inside the cube root becomes: $$27x^{-6}y^{6}$$ 4. **Rewrite with positive exponents:** $$27 \frac{y^{6}}{x^{6}} = \frac{27y^{6}}{x^{6}}$$ 5. **Rewrite the cube root expression:** $$\sqrt[3]{\frac{27y^{6}}{x^{6}}} = \frac{\sqrt[3]{27y^{6}}}{\sqrt[3]{x^{6}}}$$ 6. **Simplify cube roots:** - $$\sqrt[3]{27} = 3$$ because $$3^{3} = 27$$ - $$\sqrt[3]{y^{6}} = y^{6/3} = y^{2}$$ - $$\sqrt[3]{x^{6}} = x^{6/3} = x^{2}$$ 7. **Put it all together:** $$\frac{3y^{2}}{x^{2}}$$ 8. **Rewrite as a single cube root:** $$\sqrt[3]{\frac{27y^{6}}{x^{6}}}$$ 9. **Compare with options:** - Option 1: $$\sqrt[3]{\frac{27x^{6}}{y^{6}}}$$ (incorrect) - Option 2: $$\sqrt[3]{\frac{x^{6}}{27y^{6}}}$$ (incorrect) - Option 3: $$\sqrt[3]{\frac{81x^{2}y^{8}}{\sqrt[3]{3x^{8}y^{2}}}}$$ (incorrect and different form) - Option 4: $$\sqrt[3]{27x^{6}y^{6}}$$ (incorrect) **Answer:** The simplified form matches $$\sqrt[3]{\frac{27y^{6}}{x^{6}}}$$ which is not exactly listed, but the closest correct equivalent is option 1 if variables are swapped. Since none exactly match, the expression simplifies to $$\sqrt[3]{\frac{27y^{6}}{x^{6}}}$$. **Final simplified expression:** $$\sqrt[3]{\frac{27y^{6}}{x^{6}}}$$