1. **State the problem:** Simplify the expression $$\frac{2}{\sqrt[3]{3} - \sqrt[4]{5}}$$. 2. **Formula and rules:** To simplify expressions with radicals in the denominator, we rationalize the denominator by multiplying numerator and denominator by the conjugate or an expression that removes the radicals. 3. **Identify the conjugate:** The denominator is $$\sqrt[3]{3} - \sqrt[4]{5}$$. Its conjugate is $$\sqrt[3]{3} + \sqrt[4]{5}$$. 4. **Multiply numerator and denominator by the conjugate:** $$\frac{2}{\sqrt[3]{3} - \sqrt[4]{5}} \times \frac{\sqrt[3]{3} + \sqrt[4]{5}}{\sqrt[3]{3} + \sqrt[4]{5}} = \frac{2(\sqrt[3]{3} + \sqrt[4]{5})}{(\sqrt[3]{3})^2 - (\sqrt[4]{5})^2}$$ 5. **Simplify the denominator:** $$(\sqrt[3]{3})^2 = 3^{\frac{2}{3}}$$ $$(\sqrt[4]{5})^2 = 5^{\frac{1}{2}}$$ So denominator becomes: $$3^{\frac{2}{3}} - 5^{\frac{1}{2}}$$ 6. **Final simplified expression:** $$\frac{2(\sqrt[3]{3} + \sqrt[4]{5})}{3^{\frac{2}{3}} - 5^{\frac{1}{2}}}$$ This is the simplified form with a rationalized denominator. **Answer:** $$\boxed{\frac{2(\sqrt[3]{3} + \sqrt[4]{5})}{3^{\frac{2}{3}} - 5^{\frac{1}{2}}}}$$