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Square root

Any number can be seen as the sum or product of other numbers. For example as 25 is the sum of 16 and 9 then (16 + 9) x (16 + 9) can be used in the following fashion:

25² can be written as (16 + 9)²

Now (16 + 9) x (16 + 9) can be expanded to 16x16 + 16x9 +16x9 + 9x9 which provides the basic formula that c²=(a+b)² = a² + 2ab + b²

It is from this later expansion that the square root of a number can be calculated manually as follows:

As we know the answer to the simple value of 25² is 625 then I will root the 625 as follows, the method is to find suitable values for a and b.

The number to operated on is 625 and this number is placed similarly to a long division example under a bracket and horizontal line
______
) 625

Noting that a square root of a number must be half the digits of the number in question the number 625 is separated by a space, every two digits from the decimal point. So we write it as
___________
) 6 25.00 00

The subsequent pairs of zero are only shown as the layout where the answer may require a decimal value, a pair of zeros for each decimal place to be calculated.

The first job is to find the highest number (single digit) than when squared is less than 6, in this case it is 2. The 2 is placed above the 6
_2__________
) 6 25.00 00

The digit 2 is in effect the a in the algebraic formula. This number is squared to 4 to represent the a² and the value is subtracted from the 6

  _2__________
2 ) 6 25.00 00
    4

The next part is to find 2ab and in similar fashion.
First take away the value of the square 4 from 6 and bring down the next pair of digits

  _2__________
2 ) 6 25.00 00
    4
    4 25

The a is multiplied by 2 > 2a and placed outside as follows:

  _2__________
2 ) 6 25.00 00
    4
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