1. **State the problem:** Simplify the expression $$\sqrt[4]{2x^2} \cdot \sqrt[4]{8y^3}$$ and verify the given equality. 2. **Recall the property of radicals:** For the same root index, multiplication inside the root can be combined as $$\sqrt[4]{a} \cdot \sqrt[4]{b} = \sqrt[4]{ab}$$ 3. **Apply the property:** $$\sqrt[4]{2x^2} \cdot \sqrt[4]{8y^3} = \sqrt[4]{(2x^2)(8y^3)} = \sqrt[4]{16x^2y^3}$$ 4. **Simplify inside the root:** Since $$16 = 2^4$$, rewrite: $$\sqrt[4]{16x^2y^3} = \sqrt[4]{2^4 \cdot x^2 \cdot y^3}$$ 5. **Separate the perfect fourth power:** $$\sqrt[4]{2^4 \cdot x^2 \cdot y^3} = \sqrt[4]{2^4} \cdot \sqrt[4]{x^2} \cdot \sqrt[4]{y^3} = 2 \cdot x^{\frac{2}{4}} \cdot y^{\frac{3}{4}} = 2 \cdot x^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$$ 6. **Final simplified form:** $$2x^{\frac{1}{2}}y^{\frac{3}{4}}$$ 7. **Check the given equality:** The user wrote $$\sqrt[4]{(2^4)(x^2)(y^3)} = 2$$ which is incomplete because the variables remain under the root or as fractional exponents. **Summary:** The product simplifies to $$2x^{\frac{1}{2}}y^{\frac{3}{4}}$$, not just 2.