jmc

First, notice that all three integers have a common factor of 10. We can remove this factor from the cube root as shown: $$\begin{array}{rl}\sqrt[3]{10^3\cdot 3^3+10^3\cdot 4^3+10^3\cdot 5^3}& =\sqrt[3]{10^3(3^3+4^3+5^3)}\\ & =10\sqrt[3]{3^3+4^3+5^3}.\end{array}$$ Now, evaluate the expression under the cube root: $$10\sqrt[3]{3^3+4^3+5^3}=10\sqrt[3]{27+64+125}=10\sqrt[3]{216}.$$ Since $216=6^3$, this expression simplifies to: $$10\sqrt[3]{6^3}=\boxed{60}.$$