To find the average score, we need to calculate the total runs scored and divide it by the number of overs played.

First, let's calculate the total runs scored in the first four overs:
Total runs = 7 + 5 + 6 + 10 = 28

Next, we need to calculate the number of balls played in the first four overs:
Number of balls in 4 overs = 6 balls per over * 4 overs = 24 balls

Now, let's calculate the average score after the first four overs:
Average score = Total runs / Number of overs = 28 runs / 4 overs = 7 runs per over

Finally, we need to add the runs scored in the first three balls of the fifth over to the total runs:
Total runs = 28 + 8 = 36

Since we have played 4 overs and 3 balls of the fifth over, the number of balls played is:
Number of balls = 6 balls per over * 4 overs + 3 balls = 27 balls

Now, let's calculate the average score after the fifth over:
Average score = Total runs / Number of overs = 36 runs / 4.45 overs (approx.) = 8.09 runs per over (approx.)

Therefore, the average score after the batsman makes 8 runs in the first three balls of the fifth over is approximately 8.09 runs per over.

Please note that this calculation assumes that each ball is played and scored independently, and it does not take into account any other factors such as wickets lost or extras.