It may actually be turning on, but the display isn't. Not an uncommon thing. It may well either have a loose connection for the display or even the display could need replacing.
You may be able to (if it lets you, some devices won't show if you can't tap something on the screen) try connecting it to a computer via USB cord and see if you can access the tablet or any SD card in it. At least this way you can get your data (again if it lets you) before having anything done to it or replacing it.
I suggest contacting the manufacturer if it is still under warranty, or if it isn't, then try a local tech.