Extracting Square Roots

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Before learning the procedure, it is required that the performer memorizes the squares of the numbers 1-20 which is very elementary:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25 62 = 36
72 = 49
82 = 64
92 = 81
102 = 100 112 = 121
122 = 144
132 = 169
142 = 196
152 = 225 162 = 256
172 = 289
182 = 324
192 = 361
202 = 400

Technique – 1

To extract the square root of any perfect square, follow the below steps:

Step 1: Omit the digit next the last digit.
Step 2: Find the square root of its extremities. To do this, always start with right side. (If the right digit is not a perfect square, then start from the left extremity) Find the square root of that and write it down.
Step 3: Subtract this finding from 10. Write the result down too.
Step 4: Find the square root of left extremities. This is the tens digit of final answer.
Step 5: Write the result 2 times. Once with the square root of right digit and secondly with the number found after subtracting that from 10. Now, there are 2 possible answers one greater and one smaller; both with the same tens digit.
Step 6: To find the right answer, multiply the square root of left extremities with its successor. If left extremities are greater than the product, the right answer would be the greater option and if left extremities are less than the product, the right answer would be the smaller option. It’s so simple!

Let us illustrate the method with some examples:

Extracting square root of 441 (√441)

1) Skip the digit next the last digit which is 4 (√4x1)
2) Square root of right extremity i.e. 1 is 1
3) 10 minus 1 is 9
4) Square root of left extremity i.e. 4 is 2
5) Write down 2 as tens digit with both 1 and 9 to get 2 possible answers (21 and 29)
6) Now, 2 times its successor which is 3 (2x3) is 6. The left extremity which is 4 is less than 6. Therefore, the right answer must be the smaller option which is 21.

Let’s go to another example: √784 (square root of 784)

1) Skip the digit next the last digit which is 8 (√7x4)
2) Square root of right extremity which is 4 is 2
3) 10 minus 2 is 8
4) Square root of left extremity which is 7 is…. Well, 7 is not a perfect square. So consider the lower highest perfect square which is 4. Square root of 4 is 2.
5) Write down 2 as tens digit with both 2 and 8 to get 2 possible answers (22 and 28)
6) Now, 2 times its successor which is 3 (2x3) is 6. The left extremity which is 7 is greater than 6. Therefore, the right answer must be the greater option which is 28.

Let’s go for another example: √3969 (square root of 3969)

1) Disrgard the digit next the last digit which is 6 (√39x9)
2) Square root of right extremity which is 9 is 3
3) 10 minus 3 is 7
4) Square root of left extremities which is 39 is…. Well, 39 is not a perfect square. So consider the lower highest perfect square which is 36. Square root of 36 is 6.
5) Write down 6 as tens digit with both 3 and 7 to get 2 possible answers (63 and 67)
6) Now, 6 times its successor which is 7 (6x7) is 42. The left extremity which is 39 is smaller than 42. Therefore, the right answer must be the smaller option which is 63.

Further example: √5476 (square root of 5476)

1) Disrgard the digit next the last digit which is 7 (√54x6)
2) Square root of right extremity which is 6 is…. Well here 6 is not a perfect square so it sould be solved from the left extremities which is 54. Now, 54 is not also a perfect square. So consider the lower highest perfect square which is 49. Square root of 49 is 7.
3) Write down 7 as tens digit with the left extremitiy 6 and 4 (10-6) to get 2 possible answers (76 and 74)
4) Now, 7 times its successor which is 8 (7x8) is 56. The left extremity which is 54 is smaller than 56. Therefore, the right answer must be the smaller option which is 74.

√625 (square root of 625)

1) Disrgard the digit next the last digit which is 2 (√6x5)
2) Square root of right extremity which is 5 is…. Well here 5 is not a perfect square so it sould be solved from the left extremities which is 6. Now, 6 is not also a perfect square. So consider the lower highest perfect square which is 4. Square root of 4 is 2.
3) Write down 2 as tens digit with the left extremitiy 5 and 5 (10-5) to get 2 possible answers (25 and 25)
4) Now, 2 times its successor which is 3 (2x3) is 6. The left extremity which is 6 is equal to 6. Therefore, the right answer must be the greater option which is 25. In this case both options are same. So, the answer must be 25.


One final example: √13689 (square root of 13689)

1) Skip the digit next the last digit which is 8 (√136x9)
2) Square root of right extremity which is 9 is 3
3) 10 minus 3 is 7
4) Square root of left extremity which is 136 is…. Well, 136 is not a perfect square. So consider the lower highest perfect square which is 121. Square root of 121 is 11.
5) Write down 11 as tens digit with both 3 and 7 to get 2 possible answers (113 and 117)
6) Now, 11 times its successor which is 12 (11x12) is 132. The left extremity which is 136 is greater than 132. Therefore, the right answer must be the greater option which is 117.

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