1. The problem is to simplify the expression $\sqrt[3]{34} \times \sqrt[3]{2}$. 2. Recall the property of cube roots: $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}$. 3. Apply this property to combine the cube roots: $$\sqrt[3]{34} \times \sqrt[3]{2} = \sqrt[3]{34 \times 2}$$ 4. Multiply inside the cube root: $$34 \times 2 = 68$$ 5. So the expression simplifies to: $$\sqrt[3]{68}$$ Thus, the simplified result is $\boxed{\sqrt[3]{68}}$.