Methods to find square roots

What are the methods to find square root of imperfect square?|how to find square root by long division?| How to find square root by observation?

How to manually find a square root

1.By long division method

First, divide the number to be square-rooted into pairs of digits, starting at the decimal point. That is, no digit pair should straddle a decimal point. (For example, split 1225 into "12 25" rather than "1 22 5"; 6.5536 into "6. 55 36" rather than"6.5 53 6".)
Then you can put some lines over each digit pair, and a bar to the left, somewhat as in long division. 

     +--- ---- ----
     | 4   66   56

Find the largest number whose square is less than or equal to the leading digit pair. In this case, the leading digit pair is 4; the largest number whose square is less than or equal to 4 is 2.
Put that number on the left side, and above the first digit pair. 

       2
     +--- ---- ----
  2  | 4   66   56

Now square that number, and subtract from the leading digit pair. 

       2
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
       0

Extend the left bracket; multiply the last (and only) digit of the left-hand number by 2, put it to the left of the difference you just calculated, and leave an empty decimal place next to it. 

       2
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 4_  | 0

Then bring down the next digit pair and put it to the right of the difference. 

       2
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 4_  | 0   66

Find the largest number to put in this blank decimal place such that that number, times the number already there plus the decimal place, will be less than the current difference. For example, see if 1 * 41 is ≤ 66, then 2*42 ≤ 66, etc. In this case it's a 1. Put this number in the blank you left, and in the next decimal place on the result row on the top. 

       2    1
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 41  | 0   66

Now subtract the product you just found. 

       2    1
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 41  | 0   66
     |-    41
     +--------
           25

Now, repeat as before: Take the number in the left column (here, 41) and double its last digit (giving you 42). Copy this below in the left column, and leave a blank space next to it. (Double the last digit with carry: for example, if you had not 41 but 49, which is 40+9, you should copy down 40+18 which is 58.) Also, bring down the next digit pair on the right. 

       2    1
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 41  | 0   66
     |-    41
     +--------
42_        25   56

Now, find the largest digit (call it #) such that 42# * # ≤ 2556. Here, it turns out that 426 * 6 = 2556 exactly. 

       2    1    6
     +--- ---- ----
  2  | 4   66   56
     |-4
     +----
 41  | 0   66
     |-    41
     +--------
426  |     25   56
     |-    25   56
     +-------------
                 0

When the difference is zero, you have an exact square root and you're done. Otherwise, you can keep finding more decimal places for as long as you want.

---

Here is another example, with less annotation.

          7 .  2  8  0  1 ...
       +----------------------
7      | 53 . 00 00 00 00 00
       | 49
       +----------------------
142    |  4   00
       |  2   84
       +----------------------
1448   |  1   16 00
       |  1   15 84
       +----------------------
14560  |         16 00
       |             0
       +----------------------
145601 |         16 00 00
       |         14 56 01
       +----------------------
       |          1 43 99 00
                         ...

2 . By prime factorisation method

To find the square root of a perfect square by using 

the prime factorization method when a given number is

a perfect square

Step I: Resolve the given number into prime factors. 

Step II: Make pairs of similar factors. 

Step III: Take the product of prime factors, choosing one factor out of every pair

Resolving 484 as the product of primes, we get







484 = 2 × 2 × 11 × 11 

√484 = √(2 × 2 × 11 × 11) 

= 2 × 11

Therefore, √484 = 22  









3.By observation method: 


An interesting thing to notice is how the units place for the square changes as the units place for the number changes. This would be helpful in finding out square roots.

Step 1: First of all group the number in pairs of 2 starting from the right.

Step 2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.

Step 3: To get the unit’s place digit of the square root

Remember the following
If number ends Unit’s place digit of the 
in. square root
1. 1 or 9(10-1)
4. 2 or 8(10-2)
9. 3 or 7(10-3)
6. 4or 6(10-4)
5. 5
0. 0

Lets see the logic behind this for a better understanding
We know,
1²=1
2²=4
3²=9
4²=16
5²=25
6²=36
7²=49
8²=64
9²=81
10²=100

Now, observe the unit’s place digit of all the squares.
Do you find anything common?

We notice that,
Unit’s place digit of both 1² and 9² is 1.
Unit’s place digit of both 2² and 8² is 4
Unit’s place digit of both 3² and 7² is 9
Unit’s place digit of both 4² and 6² is 6.

Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.

Remember,
 If first pair of the original number > Result obtained on multiplication then select the greater number out of the two numbers as the unit’s place digit of the square root.

If firstpair of the original number < the result obtained on multiplication,then select the lesser number out of the two numbers as the unit’s place digit of the square root.

Let us consider an example to get a better understanding of the method

Example 1: √784=?
Step 1: We start by grouping the numbers in pairs of two from right as follows
7 84

Step 2: To get the ten’s place digit,
We find that nearest square to first group (7) is 4 and √4=2
Therefore ten’s place digit=2

Step 3: To get the unit’s place digit,
We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,
2x3=6

ten’s place digit of original number > Multiplication result
7>6

So we need to select the greater number (8) as the unit’s place digit of the square root.

Unit's place digit =8

Ans:√784=28