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<sqrt(N)< 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

`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)<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`

**Further Reading**