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My HP48SX had not been powered on in at least ten years; possibly more. It was in its plastic pouch, in a storage box, up in some closet in my apartment. I just took it down, put fresh batteries in, but it will not power up. I know the batteries are making good contact, I can measure the voltage across the actual contacts. No combination of ON, whether by itself or with other keys will wake it up, though. What can possibly have failed?

It's not that I really need it, but I am curious as to what could have gone wrong with the thing just sitting in a box in an apartment.

Try applying pressure on the bezel between the LCD and the top row of keys while trying to use the On key. If that works then the foam that holds the keyboard flex PCB to the main board has dried up. This can affect more than just the On key so you may need to apply pressure there to operate the calculator. Both of my Pioneers (a HP-20S and a HP-22S) and my HP-48SX have a similar issue. It can be fixed but it is not easy.