Will the AC power cable of a ps2 work on a slim ps3?

or do I need to buy one?

No, the PS3 is a 3 prong outlet cable, and the PS2 is 2 prong. Also the end that plugs into the console is different for both. You can use a PC power cord for the PS3 if needed, the ends are the same.

Yes it's a standard 3 prong pc plug.