In this page, finding square root using long division method, is introduced in a simple thought process.

*click on the content to continue..*

What is the `sqrt(8)`?

- `4`
- `2 sqrt(2)`
- `2 sqrt(2)`

The answer is "`2 sqrt(2)`".

The prime factorization is given as

`sqrt(8)`

`=sqrt(2xx2xx2)`

`=2xxsqrt(2)`

The prime factorization method is suitable for square roots resulting in integer values.

Square root of a number can be given as

`x^2=(10a+b)^2`

`x^2=100a^2+(20ab)+b^2`

Note:

`b^2` is a `2` digit number with tens-units places

`100a^2` is a number that has `00` at tens-units places

This understanding gives a method to eliminate `b` and look at `a` to choose the highest digit of the square root.

`x^2=color(coral)(100a^2)+color(deepskyblue)((20a+b) xx b)`

`x^2 color(coral)(- 100a^2)= color(deepskyblue)((20a+b) xx b)`

The rearranged one with `(20a+b)` gives the method to multiply `a` by `2` (which is `2a`) and append a value `b`, which is `20a+b`. Then multiply `b` to `20a+b`.*step 1 *

`x^2 color(coral)(- a^2 text( at 100s place)) = y`*step 2 *

`y - color(deepskyblue)((2a text( joined with ) b) xx b)`

In this process, the choice of `a` and `b` make the square root `x=10a+b = a text( joined with )b`

The above process is explained for 2 digit square root and is easily extended for higher number of digits.

`529`

`=23^2`

`=(10xx2+3)^2`

`=(10xx2)^2+ 2 xx 10xx2 xx 3+3^2`

`=color(coral)(100xx2^2)+color(deepskyblue)((2xx2xx10 + 3)xx3)`

`=color(coral)(2^2 text( 100s place ))+color(deepskyblue)((2xx2 text( joined )3 = 43)xx 3)`*The above steps is used in reverse when the square root is not known *

`sqrt(529)`

`=color(coral)(2) text( at 10s place ) + color(deepskyblue)(3) text( at units place)`

`=23`

This procedure is illustrated in the figure. The number is split as `5,29`.

Consider `5` first and choose `2` as the first digit.

`5-2xx2 = 1`

The first step is completed with `2`.

Then `129` is considered and the first digit `2` multiplied `2` is `4`. This `4` is the tens position and second digit `3` is chosen.

`43 xx 3 = 129`.

The square root of `529` is `23`.

*Solved Exercise Problem: *

What is `sqrt(69169)`?

- `263`
- `263`
- `11`

The answer is "`263`"

**Procedure to finding Square Root of a number** : Long division method is illustrated in the figure.

*slide-show version coming soon*