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Stephen Hawking proposed that a black hole emits radiation as it evaporates.

Black holes don't let any radiation escape their event horizon because even particles with the speed of light get trapped by their magnetic field.

Does that mean when black holes emit radiation, they just re-absorb it?

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The black hole itself does not emit the radiation, but rather the space just around it. Because of quantum effects, small particles and anti-particles will spontaneously create everywhere (and later annihilate themselves in a collision). If this happens just outside the event horizon, one particle will inevitably fall into it, never to be seen again. The other particle is now 'free' and this is the radiation Hawking predicted.

Can we detect a black hole's radiation?

Well, this radiation is way too small to be detected by itself, even if the black hole would be at our doorstep. However, the radiation makes the black hole less massive and smaller and finally it will vanish, producing a gamma ray flash which we hope is detectable, even from a very large distance.
As @JamesK notes in the comments, this evaporation will take extremely long for the 'common' black holes caused by supernovae and those found in galaxy centers, but smaller primordial black holes, formed in the very beginning of the universe, could be evaporating now and that's why we have a telescope looking for those gamma ray flashes.

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    $\begingroup$ Could note that a regular solar sized black hole could not evaporate in the 13 billion years since the start of the universe, but primordial black holes could be much smaller, and so could be evaporating now. $\endgroup$ – James K Mar 19 '18 at 19:41
  • $\begingroup$ Testing, one two. Hawking's black hole explosions? paper is worth a read. $\endgroup$ – John Duffield Apr 30 '18 at 21:18

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