If a piano tuner hears 3 beats each second when listening to a 8 Hz tuning fork and a single string on the piano which is of higher frequenc?

The short answer is that there are two possibilities:

(i) The string frequency could be 5 Hz. In this case the beat frequency would be (8 - 5) = 3 Hz

OR

(ii) The string frequency could be 11 Hz. In this case the beat frequency would be (11 - 8) = 3 Hz

In a real-life scenario, away from this nonsense, the way to find out which it was would be to stick a small piece of gum or something on to one of the prongs of the tuning fork. That would lower the frequency of the fork slightly to something a bit less than 8 Hz.

Listen again. If the answer was option (i) the beat frequency would be

( [A bit less than 8] - 5) = A bit less than 3 Hz

If it was really option (ii), the new beat frequency would be

( 11 - [A bit less than 8] ) = A bit more than 3 Hz

So you'd know which one it was.

1) 8 Hz tuning forks don't exist (if they did, they'd require a semi for transport)

2) human hearing, at the most, goes down to about 20 Hz

3) a regular piano goes down to about 27.5 Hz, very few large concert grand pianos can reach down to 16.35 Hz

So

a) the piano string will have the higher frequency in every case

b) n*8 + 3, the lowest possibility being 19 Hz since there are no piano strings even remotely close to 11 Hz, quite apart from the fact that noone would be able to hear them