Question

Between what two consecutive integers do `sqrt310` lie?

Answer 1

Between `17` and `18`.

Explanation:

I'd say that the simpliest way is just to go and try, of course with a little bit of wisdom.
For example, we know (or easily compute) that `10^2=100`, and `20^2=400`. So, `sqrt(310)` will surely be between `10` and `20`.

Now we narrow our search by confronting with midpoints: `15^2=225`, so `sqrt(310)` is between `15` and `20`.

Since `17^2=289`, `sqrt(310)` must be between `17` and `20`

But `18^2=324`, so `sqrt(310)` is exactly between `17` and `18`.

Of course, I solved the exercise using only confrontations and easy computing. One immediate way is simply to ask any calculator for `sqrt(310)`, he will give you back `17.6068...` and you'll immediately get the solution.