Another New Fastest Method of Finding Sqrt, Non Sqrt Roots.

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Introducing The Most Fastest Way to Find Perfect Square Roots.

Creating Successive Series Beforehand.
0 -- 1 has zero difference.
1 --- 3 has only as difference of one.
3 ---- 6 has a difference of two.
6 ---- 10 has a difference of three.
10 ---- 15 has a difference of four.
.
.
.

This series is a form of infinite successive series where integers at the above right hand side, that is 1, 3, 6, 10 15…. (highlighted below in yellow color) is the result of addition of positive integers in following sequence such as 2, 3, 4, 5, 6, 7, 8….(highlighted below in green color)

0 -- 1
+ 2
1 --- 3
+ 3
3 ---- 6
+ 4
6 ---- 10
+ 5
10 ---- 15

While finding square roots solutions, the use of successive series is to simply avoid calculation timing by creating it beforehand.
Therefore, while solving the problems of finding any square, non square or any higher roots we require this series beforehand.

eg., when we are about to find the square root of 40000.
40000/ 72 = 555
so, we need check where the 555 appears at the successive series.Therefore, we need a ready successive series.
So, as shown above, one can continue go on creating same series by adding 15 + 6 = 21, 21 + 7 = 28 ….. keep record of it and use it whenever required to calculate key integer ‘c' of any other problems of finding square roots.

Example Solution -
To find perfect square root of √3249

1st Step - Divide 3249 by 72 to get m.

3249 / 72 = 45.125 …..
By ignoring the decimals, we get m = 45

2nd Step – Finding the Key Integer c .

0 -- 1 has zero difference.
1 --- 3 as difference of one.
3 ---- 6 has a difference of two.
6 ---- 10 has a difference of three.
10 ---- 15 has a difference of four .
.
.
45 ---- 55 has a difference of nine.
( Note - Just for presentation purpose, after 10 ---- 15 I have directly shown the ninth step of series. One can continue calculate and check until 45 ---- 55).

Now, from 1st step, we have m = 45
As per the above successive series, m appears at 45 to 55 and the series has a difference of nine therefore, we can take c = 9


3rd Step – Multiplying c by 6.
From 2nd step, we have c = 9
Therefore, 9 × 6 = 54
Now we get c = 54
Checking whether c is final answer by dividing 3249 by 54 . We found it is not divisible.
Therefore, we will proceed to below 4th step of checking rules.

4th Step – Checking the Rules of Finding ‘c’ .
√X =3249 is divisible integer divisible by 3. Therefore it follows the fourth rule of finding c.
Fourth Rule - If the √ X is any odd integer and is divisible by 3, then the final answer c will be adding c by 3 i.e. ‘c + 3'.
54 + 3 = 57
Since 3249 is divisible by 57
Therefore, √3249 = 57
Note - Rules are available at the paper, check below attachment. Also the attachment has many more examples of non square roots, cube and higher roots.