|If current times voltage equals power, this makes it seem that high current times low voltage would equal low current times high voltage; but this is not true because of resistance. How is resistance taken into account in the current times voltage equal power equation?
Your first observation, that high current times low voltage would equal low current times high voltage is true; it means that electricity can deliver the same power in two different ways: as a large current of low energy charges or as a small current of high energy charges. That result is critical to the electrical power distribution system. The resistance problem is a side issue: it makes the delivery of power as a large current of low energy charges difficult. If you could get this current to peoples' houses without wasting its power, there would be no problem, but that delivery isn't easy. The wires waste lots of power when you try to deliver these large currents. So the electric power distribution system uses small currents of high-energy charges instead.