It's like the car in Back to the Future 2. They convert the organic material in to energy which powers them.
Its not really clear in the game or grimoire, but it's likely done in a cold fusion process, since the amount of power needed to simulate the computing power of a human brain as well as move the exo's heavy metal body and limbs would be astronomical.
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Golden Age alloys probably cut down on the weight significantly. Exos appear to be human sized. All that weight in relatively small frames would give them a lot of mass if they used metals known to us today.
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Still would take a lot of power to actuate. CDs aren't relatively heavy but CD players used to kill batteries like crazy because it took a lot of power to spin the disk.
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You're not thinking Golden Age tech though.
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Yeah, but CDs aren't super sense or heavy. Even with new alloys and tech, things still have to be made of substances from our physical universe. I'm power and battery tech is way more advanced though. An exo's brain is essentially a digital human brain. That kind of processing would take way more power than actuating the exo's body parts, whether they're heavy or light weight.