Variations of MSM-2

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

Rule 2: If the given problem ends with …025

1) 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

1) N = 2 and 7

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

Case No. 2

Given:

√66,341,025

Step 1:

(Left Column, 1^st Row)

√66’34’10’25

... 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)

1^st Square Root Locator

.. 72’25

.. 68’06 \ ↓

.. 64..... /

.136’25 / 2

.. 68’12

Modify Step 2:

P : 66’34

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)

2^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)

3^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

8245 ← Eliminated

8195 ← Eliminated

8155

8145

Step 7:

(Left Column, 4^th Row)

4^th Sq. Rt. Loc.

↓ 8145 ........ 8155 ↑

.. 67’04

.. 66’53 \ ↓

.. 66’03 /

.133’07/ 2

.. 66’53

Step 8:

(Right Column, 4^th Row)

Therefore:

√66,341,025 = 8,145

Case No. 3

Given:

√54,390,625

Step 1:

(Left Column, 1^st Row)

√54’39’06’25

....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

Step 3:

(Left Column, 2^nd Row)

1^st Square Root Locator

.. 56’25 \

.. 52’56 / ↑

.. 49 .....

.105’25 / 2

.. 52’62

Modify Step 2:

P : 54’39

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)

2^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

We can separate them into two groups:

.. ↑

7375

7325

.. ↓

7325

7275

Step 6:

(Right Column, 3^rd Row)

3^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)

4^th Sq. Rt. Loc.

↓7325 ......... 7375↑

.. 54’40 \

.. 53’94 / ↑

.. 53’48

.107’88 / 2

.. 53’94

The ‘up arrow’ indicates that the ‘true’ square root is 7375

Step 8:

(Right Column, 4^th Row)

Therefore:

√54,390,625 = 7,375