Toshiba laptop won't boot

Hi,

I have a Toshiba Satellite Pro C660-2JQ laptop which won't boot. The problem started after it had accidentally been on sleep for about 3 days: I opened the lid to wake it up and it just switched off. When I try to switch it on, the power light comes on and the fan whirs for a few seconds, then nothing. I sent it back to the company I got it from and (sod's law) it switched on fine. After stress testing it, they thought the problem might have been with the RAM. They fixed it (I think by moving it but I know nothing about computers and their jargon-heavy explanation went right over my head).

The laptop worked fine for a couple of days but the problem has resurfaced. I've tried holding the power button down with the battery out and switching it on running on the mains.

Any help would be greatly appreciated! I'd like to keep my data, so would rather not reset the laptop.

Thank you