Black holes emit radiation (Hawking radiation) and sometimes even gamma radiation (when "eating" a star), but nothing can escape from a black hole. So the black hole should "eat" its own radiation. But of course it emits radiation, so whats going on?
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Radiation "from a black hole" is actually radiation that arises around a black hole.
As matter is funneled in towards the event horizon, it is compressed and becomes hot. Conservation of angular momentum leads to the formation an accretion disk.
The observed radiation comes from this accretion disk, not from the black hole itself. The inner edge of the accretion disk is often taken to be the innermost stable circular orbit at (approximately) three Schwarzschild radii.
EDIT: I was assuming you were asking about black holes that are actually observed. These are massive enough (a few solar masses and above) that Hawking radiation is totally insignificant. If you are asking about hypothetical mini- black holes, then even in that case the Hawking radiation arises from outside the event horizon.
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$\begingroup$ I would only add that Hawking radiation does come from the black hole itself but is created just outside the horizon, so that it can escape if it has enough outward momentum. $\endgroup$– NoiralefCommented Feb 9, 2017 at 8:14