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.
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
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))`
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`
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)<sqrt(30)<(241/44)`
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`