Question 120529
Look at a table of squares:


{{{matrix(9,2,
1,1,
2,4,
3,9,
4,16,
5,25,
6,36,
7,49,
8,64,
9,81
)}}}

14 falls in the interval {{{9<14<16}}}, therefore {{{sqrt(14)}}} falls in the interval {{{3<sqrt(14)<4}}}.


The difference between 9 and 16 is 7, and the difference between 9 and 14 is 5, so 14 lies 5/7ths of the way along the interval from 9 to 16, so {{{sqrt(14)}}} should lie approximately 5/7ths of the way along the interval from 3 to 4, in other words, 3 and 5/7ths.


3 and 5/7ths, or 26/7ths rounds to 3.71.  The calculator value for {{{sqrt(14)}}} rounds to 3.74.  Close enough.


Hope that helps,
John