This video shows the correct procedure to dismantle a first generation iPod Nano. The presenter starts by using a putty knife (aka the iPod take apart tool) and separates the casing. Next, the presenter removes the battery from the internal circuit board by using a soldering iron to melt the soldered battery connections. A small Phillips head screw driver is then used to remove three screws from the logic board. Finally removing two small ribbon cables will free the logic board from the plastic casing. Now you can remove the LCD display as well.
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