Also, the power can be expressed as P=E*I and with a little algebra we can combined these expressions and derive P=EČ / R
So what does all this mean? Well, for one thing it becomes clear that an appliance (load) that draws 1 amp (ampere) of current at 120 volts will draw 10 times as much current at 12 volts (1/10 the voltage) or 10 amps. Since P=E*I then 120 volts times 1 amp = 120 watts. Also, 12 volts times 10 amps = 120 watts. So you can see that the power remains the same. As the Voltage goes down, the Amperage increases to maintain the power which will be determined by the 3rd factor, resistance.
Ok, now let's say you have a nice 1200 watt hairdryer. Well, that would work out to 10 amps at 120 volts. But, when your power inverter uses the 12 volts supplied from your batteries, the amperage goes up to 100 amps to produce the same 1200 watts! (P=E*I). This means that even the very large cables connecting your batteries to the inverter will get warm. This is why it becomes impractical or impossible to run say, a 4000 watt electric clothes dryer. Even if you had large enough wires to handle the required 333 or so amps, your batteries would not last long.
It is true that the cables will not get as warm if the current can be reduced by increasing the voltage by using a 24 volt battery system or even a 48 volt battery system. This still will not change the amount of power that your batteries must supply.