What you have
What you have is a MacBook Pro. The PowerBook brand was discontinued around 2005 when they went with an all "Mac" branding scheme.
If this is a 2007-2008 non-unibody MacBook Pro, take the thing back to the Apple store or some AASP and tell them to run the nVidia graphics chip test on it. It may be a bit of a long shot in your case, but if it pans out, it's a free logic board replacement. If they have no idea what you're talking about, take your laptop somewhere else.
I'm becoming less and less impressed with Apple store "Genius" diagnoses. Just fixed an iMac today where they said it was a bad CPU, but all it was, was a bad memory module. Had they simply run some EFI level memory tests they would have seen it fail one after the other. So I fixed the thing for around $200US where the Apple store diagnosis would have been more like $1400US. Do not hesitate to take it somewhere else for a second opinion. I'm sure they have some good retail level techs, but it's really no different from your average Best Buy or Fry's store tech. It's a crapshoot whether or not you'll get a good one, and honestly, the good ones usually don't stick around very long because they have the skills to get a better job. With Apple stuff, I'd honestly look for an AASP, where they actually need to earn your business a lot more than an Apple retail store. Which isn't to say there aren't plenty of shady operations around, but plenty of smaller AASP repair shops will likely hire people based more on their technical competency rather than social skills like Apple does for its retail stores.