1. **State the problem:** We need to find the length $z$ of a cube given its volume is 729 mm$^3$. 2. **Formula used:** The volume $V$ of a cube with side length $z$ is given by: $$V = z^3$$ 3. **Apply the formula:** Substitute the given volume: $$729 = z^3$$ 4. **Solve for $z$:** To find $z$, take the cube root of both sides: $$z = \sqrt[3]{729}$$ 5. **Calculate the cube root:** Since $729 = 9^3$, we have: $$z = 9$$ 6. **Answer:** The length $z$ of the cube is 9 mm. This means each edge of the cube measures 9 mm to have a volume of 729 mm$^3$.