1. **State the problem:** Rationalize the denominator of the fraction $\frac{3}{\sqrt[3]{2}}$. 2. **Recall the formula and rule:** To rationalize a denominator with a cube root, multiply numerator and denominator by the appropriate expression to make the denominator a perfect cube. 3. **Identify the needed factor:** Since the denominator is $\sqrt[3]{2}$, multiplying by $\sqrt[3]{4}$ (because $2 \times 4 = 8$ and $\sqrt[3]{8} = 2$) will rationalize it. 4. **Multiply numerator and denominator:** $$\frac{3}{\sqrt[3]{2}} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}} = \frac{3 \sqrt[3]{4}}{\sqrt[3]{2} \times \sqrt[3]{4}}$$ 5. **Simplify denominator:** $$\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{2 \times 4} = \sqrt[3]{8} = 2$$ 6. **Final expression:** $$\frac{3 \sqrt[3]{4}}{2}$$ **Answer:** $\boxed{\frac{3 \sqrt[3]{4}}{2}}$