How can we find square root of a number by hand?

To be able to evaluate this, recall that  with a square root of N such that N= AB , it follows square root of N lies in between the A and B.
A This  is our clue that we can apply factoring for the value inside the square root sign.
For a radical root(n)(x) ,  the parts are called:
n = index
x= radicand or value inside the radical sign.
 A square root has an index of 2 which has a radical sign root(2)(x) or  sqrt(x).
 Suppose we have sqrt( 30).
Apply factoring on the radicand: 30 =5*6 then we know that
5 ltsqrt(30)lt6
To solve it numerically, note that an average of two number (A+B)/2 will be in between A and B.
 Then,  Alt(A+B)/2 ltB .
Average value= (5+6)/2 =11/2 or 5.5
 
Apply the average value to the factoring of the radicand such that:
radicand= (A+B)/2 * radicand/((A+B)/2)
Divide radicand by the average value:
sqrt(x)/(((A+B)/2)) = 30/ ((11/2))
          = 30*(2/11)
          = 60/11
Then, factoring of the radicand: 30 = 11/2* 60/11
and it follows it square root will lie in between: 
60/11 lt sqrt(30)lt11/2
Note:60/11 lt11/2  since 60/11~~5.455 and 11/2=5.5
Note that the boundary values is approximately same as "5.5" then we can estimate the value of the square root: sqrt(30)~~5.5
For more accurate estimation, repeat the same procedure with the new set of factors of the radicand:
30 = 11/2* 60/11
Then,
average value= (11/2+ 60/11)/2 = 241/44 or 5.477 rounded off.
sqrt(x)/(((A+B)/2)) =30/((241/44)) = 1320/241
then new factoring: 30=(241/44)*(1320/241)
new range will be:(1320/241)or 5.47718 ltsqrt(30)lt5.477273
Note that the boundary values is approximately same as "5.4772" then we can estimate the value of the square root:  sqrt(30)~~5.4772
http://mathforum.org/library/drmath/view/52638.html