Three solid metal spheres of radii 3 cm, 4 cm and 5 cm are melted together and recast into a single solid sphere. What is the radius of the new sphere, in cm?

Volumes add, and each is (4/3)π$r^3$, so the (4/3)π cancels and it is the CUBES of the radii that add: $3^3+4^3+5^3=27+64+125=216$, giving $R$ = ∛216 = 6 cm. The trap 12 adds the radii themselves, which would be right only if volume were proportional to $r$; 4 averages them; 216 is the sum of the cubes with the cube root never taken.