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Unplug the TV's power cord, at the outlet.
Then unplug it at the other end.
Examine both ends for corrosion on the metal, or other dirt. Brush out with an old toothbrush (used dry) to make sure. Then re-connect the power cord to the TV, then plug it in at the other end also.
(If that does not help, test the outlet, by plugging some other device, e.g. a lamp, into it as a test, to make sure it is working.)
If, when you switch On the TV, it still has the power light come On, but go Off after a bit, make sure by watching it, that it is not blinking. Blink codes of Power lights on TVs can be important information about the TV and any fault it has self-detected.
How old is your RCA TV ?