1. **State the problem:** Find the constant of proportionality $k$ given that $z$ is directly proportional to the product of $x$ and the cube root of $y$, and when $x=2$, $y=8$, $z=12$. 2. **Write the formula:** Since $z$ is directly proportional to $x$ and the cube root of $y$, we write: $$z = k x \sqrt[3]{y}$$ 3. **Substitute the known values:** Given $x=2$, $y=8$, and $z=12$, substitute these into the formula: $$12 = k \times 2 \times \sqrt[3]{8}$$ 4. **Calculate the cube root:** $$\sqrt[3]{8} = 2$$ 5. **Simplify the equation:** $$12 = k \times 2 \times 2 = 4k$$ 6. **Solve for $k$:** $$k = \frac{12}{4} = 3$$ **Final answer:** $$k = 3$$