When the Hoover Dame was completed and the reservoir behind it filled with water, did the moment of inertia of the Earth increase, decrease, or stay the same?
Increased by a small amount
The definition of moment of inertia is
A dam basically lifts water up above the river bed. Hence a chunk of mass otherwise on the surface of earth has moved up further away from the center of the earth, increasing it's moment of inertia. If fact, you can estimate the increase if you look up the water volume of the dam and the average height of the water level.