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Turn the calculator over. Look for some small screws that fasten a 1 or cover to the body of the calculator. Loosen the screws and remove cover. You should find button cell batteries. Before removing them make a note of the sign you see on the batteries (+ or -) while they still in place. Replace the batteries with the same model or equivalent ones. Replace the cover.
For more detailed instructions consult your calculator's manual.
It should be marked on the calculator body or inside the battery compartment cover. In my experience the positive side has always been up. But who can be sure. Besides you do not even say what model and make.
The 48GX runs on three AAA type alkaline batteries. However, if the device was unused that long, the button cells used for backing up the RAM cards may also be depleted, and the calculator will report "low battery" displaying the ((.)) symbol, even with fresh AAA cells installed.
The RAM card mounted batteries are type CR2016.
Remove the calculator battery compartment door by pressing down and sliding it off away from the calculator.
Turn the calculator and shake the batteries out.
Replace with a fresh set of AAA alkaline batteries. Polarity is marked on the bottom of the battery compartment.
Replace the battery compartment door by sliding it back in place.
RAM card batteries:
Turn the calculator over and remove the plastic cover over the plug-in card ports (display-end of the calculator).
HP recommends to turn on the calculator while replacing the RAM card battery to avoid memory loss. I assume, your HP48GX has no marbles to recover by now anyway, so it does not matter, I guess.
Near the left side of the end of the card. you'll find the nail grip in the black plastic battery holder. Use a thumbnail to gently pull out the battery holder. There is one battery in each card (additional cards are options).
Remove the old battery from the holder and replace by a new CR2016 button cell. The "+" side of the battery should point into direction of the calculator's front.
Re-insert the holder with the new battery and check the RAM card is properly seated still.
Close the compartment by sliding back in the cover.
As long as there is a small metallic part fixed in calculator body you can find a workaround. Cut a piece of aluminum foil or other thin sheet of metal that you can easily fold and stuff it between the battery and the remnant of the coil. Do not let it touch neighboring batteries. If the broken coil has enough wire left in it you can also stretch it a bit to improve contact.
Push down the tab and lift off the battery cover on the back of the calculator.
To replace the four AAA batteries, simply pull them out and put in new ones, following the polarity indicators. To replace the backup battery, unscrew the cover, pull out the battery, put in the new one, and replace the cover.
Take out the AAA batteries and hold the on/off key for thirty seconds. Replace the batteries and it should come on. If it doesn't remove the backup battery and the AAA batteries and hold the on/off key for another 30 seconds. Replace all batteries and turn it on again.