did it shut down because it ran out of power or for some other reason that you dont know in the middle of something or did you shut it down normally and it just wont come back on.
here is the thing. there are really 2 possibilities.
1 there is a physical problem and it is going to need repair.
2. its not getting power and the battery ran out and its not charging.
so my suggestions are going to seem simplistic and maybe insulting but the only thing that i can fix is if its not getting power.
try a different outlet.
make sure the charger is connected on both sides and if it has the box in the middle that the cable to that is secure as well.
there should be an led on the charger itself. if that is not on the power outlet is bad or the charger is broken
these are asus instructions to try. pretty much what i have said but they include one for pressing a button in the reset hole. you can unfold a paper clip to do that.
1: Unable to power on the notebook.
1.1: Reset hole available on the bottom?
1.1.1: Reset hole available. --> Use a spiculate pin to tip the reset hole.
184.108.40.206: Problem solved. --> Able to power on.
220.127.116.11: Problem persists. -->1.2
1.1.2: Reset hole unavailable. --> 1.2
1.2: Try to power on just using battery.
1.2.1: Able to power on. --> 1.3
1.2.2: Unable to power on. --> Check the adapter indicator.
18.104.22.168: Adapter indicator on. --> Try to power on just using adapter.
22.214.171.124.1: Able to power on. --> 1.3
126.96.36.199.2: Unable to power on. --> Contact ASUS service center for help.
188.8.131.52: Adapter indicator off. --> Adapter problem, check the cable or contact ASUS service center for help.
1.3: Try to enter the BIOS by pressing F2 button when ASUS logo appears.
1.3.1: Able to enter the BIOS. --> 1.4
1.3.2: Unable to enter the BIOS. --> Contact ASUS service center for help.
1.4: Try to enter the Operation System.
1.4.1: Able to enter the OS. --> Reinstall the OS for a try.
184.108.40.206: Problem solved. --> Able to power on.
220.127.116.11: Problem persists. --> Contact ASUS service center for help.
1.4.2: Unable to enter the OS. --> Contact ASUS service center for help.
the post i saw suggested that the charger had a black box in he middle of it and that box had an led light to show it was working.
you might try plugging it into a different outlet to see.
it should still be under warranty but i think that i would try taking it to someplace that has another charger just to see if that is the problem before i bothered sending it back for repair. But i should be able to get you the number for asus waranty repair if you like.
that sounds like a physical problem especially if it isnt even getting a power light. sometimes its as simple as the little thing the charger plugs into broke right inside the case.
this page will give you service centers near you by zip code
i believe they also have a service where you mail it back to them and probably get a reconditioned one sent back to you immediately. but you can call this number to see what your options are that way