1. **State the problem:** We want to find an equivalent form of $$\sqrt[3]{54x^{5}y^{12}}$$ from the given options. 2. **Recall the cube root property:** $$\sqrt[3]{a^3} = a$$ and $$\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. 3. **Factor inside the cube root:** $$54x^{5}y^{12} = 54 \cdot x^{5} \cdot y^{12}$$ 4. **Prime factorize 54:** $$54 = 27 \times 2 = 3^3 \times 2$$ 5. **Rewrite the expression:** $$\sqrt[3]{3^3 \times 2 \times x^{5} \times y^{12}} = \sqrt[3]{3^3} \times \sqrt[3]{2} \times \sqrt[3]{x^{5}} \times \sqrt[3]{y^{12}}$$ 6. **Simplify each cube root:** - $$\sqrt[3]{3^3} = 3$$ - $$\sqrt[3]{2}$$ remains as is - $$\sqrt[3]{x^{5}} = x^{\frac{5}{3}} = x^{1 + \frac{2}{3}} = x \cdot \sqrt[3]{x^{2}}$$ - $$\sqrt[3]{y^{12}} = y^{\frac{12}{3}} = y^{4}$$ 7. **Combine the simplified parts:** $$3 \times y^{4} \times x \times \sqrt[3]{2 \times x^{2}} = 3xy^{4} \sqrt[3]{2x^{2}}$$ 8. **Compare with options:** Option A is $$3xy^{4} \sqrt[3]{2x^{2}}$$ which matches our result. **Final answer:** Option A