Initially, the S.E. “Grouping” Method seems to be difficult to grasp (or to understand). But the only tricky part is the

__creation of the sets of possible square roots__

__Rule 1:__ If the given problem ends with **… 225 **

**1) N = 1, 3, 6 and 8**

2) There are always two sets (or **2 groups**), each having **five (5) elements**

**… 025**

**N = 4, 5 and 9**

2) There are always two sets (or **2 groups**), each having **four (4) elements**

__Rule 3:__ If the given problem ends with **… 625 **

**N = 2 and 7**

2) There is always one set (or **1 group**), having **five (5) elements
**

__
__

__Case No. 2__

__Step 1:__

(Left Column, 1^{st} Row)

**√66’34’1 0’25**

...** 8 **...........** 5**

**N : 4, 5, 9**

__Step 2:__

(Right Column, 1^{st} Row)

**P : 66’34**

**85 : 72’25 ↓**

**4↓ M : 0, 1, 2, 3, 4**

__Step 3:__

(Left Column, 2^{nd} Row)

^{st} Square

.. **72’25**

.. **68’06 \ ↓**

__.. __64..... /

.__136’25 /__ 2

.. **68’12**

__Modify Step 2__:

85 : 72’25** ↓ /8250/**

4↓** M : 0, 1, 2, 3, 4**

__Step 4__:

List down the two sets of possible square roots

(Right Column, 2^{nd} Row)

**↓** .............. **↑**

**8145** ......... **8245**

**8095** ......... **8195**

**8055** ......... **8155**

**8045** ......... **8145**

__Step 5__:

(Left Column, 3^{rd} Row)

^{nd} Sq. Rt. Locator

.. **68’06 \**

.. **66’03 / ↑**

__.. __64 .....

.__132’06 /__ 2

.. **66’03**

__Modify Step 4__:

As an option, draw a square, enclosing the right set of possible square roots

(Right Column, 2^{nd} Row)

**8245 \ ↑**

**8195 / **

**8155 \**

**8145 / ↓**

__Step 6: __

(Right Column, 3^{rd} Row)

^{rd} Sq. Rt. Loc.

.. **68’06**

.. **67’04 \ ↓**

.. __66’03 /__

.__134’09 /__ 2

.. **67’04**

__Modify Step 4__:

As option, cross out 8245 and 8195, leaving 8155 and 8145

8195 ← Eliminated

**8155 **

**8145 **

__Step 7: __

(Left Column, 4^{th} Row)

^{th} Sq. Rt. Loc.

**↓ 8145** ........ **8155 ↑ **

..

**67’04**

..** 66’53 \ ↓**

..** 66’03 /**

.__133’07/__ 2

..** 66’53**

__Step 8:__

(Right Column, 4^{th} Row)

**√66,341,025 = 8,145
**

__
__

__Case No. 3__

Given:

**√54,390,625**

__Step 1:__

(Left Column, 1^{st} Row)

**√54’39’0 6’25**

....**7** ........... **5**

**N : 2, 7**

__Step 2:__

(Right Column, 1^{st} Row)

**P : 54’39**

**75 : 56’25 ↓**

**4↓ M : 0, 1, 2, 3, 4**

(Left Column, 2^{nd} Row)

^{st} Square

.. **56’25 \**

.. **52’56 / ↑**

.. __49 .....__

__. 105’25 /__

**2**

.. **52’62**

__Modify Step 2__:

75 : 56’25** ↓ /7250/**

**4↓ M : 0, 1, 2, 3, 4**

__Step 4__:

List down the set of possible square roots

(Right Column, 2^{nd} Row)

**7475**

**7425**

**7375**

**7325**

**7275**

We can re-group them as follows:

**↑**

**7475**

**7425**

**7375**

**↓**

**7375**

**7325**

**7275**

__Step 5__:

(Left Column, 3^{rd} Row)

^{nd} Sq. Rt. Locator

.. **56’25**

.. **54’40 \ ↓**

.. __52’56 /__

.__108’81 /__ 2

..** 54’40**

__Modify Step 4:__

The group with an up arrow is eliminated.

(As an optional, draw a square and enclose 7375, 7325, 7325)

__7375__

__7325__

__7275__

**↑**

**7375**

**7325**

**↓**

**7325**

**7275**

__Step 6: __

(Right Column, 3^{rd} Row)

^{rd} Sq. Rt. Loc.

.. **54’40 \**

..** 53’48 / ↑**

.. __52’56 __

.__106’96__/ 2

.. **53’48**

**7275 **is eliminated

__Step 7: __

(Left Column, 4^{th} Row)

^{th} Sq. Rt. Loc.

**↓7325** ......... **7375↑**

**54’40 \**

.. **53’94 / ↑**

.. __53’48 __

.__107’88 __/ 2

.. **53’94**

__Step 8:__

(Right Column, 4^{th} Row)

**
**

**
**

**√54,390,625 = 7,375**

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