One possible way is as follows:
Try something obvious to start with
2^2 = 4
3^2 = 9
so the answer is between 2 and 3
Now try the midpoint of these (2+3)/2 = 2.5
2.5^2 = 6.25
This is too low so try half-way between this and the previous high value (2.5 + 3)/2 = 2.75
2.75^2 = 7.563
This is too high so try half-way between this and the previous low value (2.75+2.5)/2 = 2.625
2.625^2 = 6.891
... and so on until you get sufficiently close.
One possible way is as follows:
Try something obvious to start with
2^2 = 4
3^2 = 9
so the answer is between 2 and 3
Now try the midpoint of these (2+3)/2 = 2.5
2.5^2 = 6.25
This is too low so try half-way between this and the previous high value (2.5 + 3)/2 = 2.75
2.75^2 = 7.563
This is too high so try half-way between this and the previous low value (2.75+2.5)/2 = 2.625
2.625^2 = 6.891
... and so on until you get sufficiently close.