As an example, let's take the square root of 819531.8784.
1. To set up the problem, put the number of which you want to find
the square root inside the square root symbol. Then, starting at
the decimal point, group the digits in pairs. Pad any leftover
single digits
with zeros.
---------------
V 81 95 31.87 84
2. Determine the highest perfect square that is less than or equal to
the first pair of digits. Put its square root above the pair of
digits, where the "quotient" would be in a long-division
problem.
9
---------------
V 81 95 31.87 84
3. Multiply this digit by itself, and subtract from the first pair of
digits. Then bring down the next pair of digits.
9
---------------
V 81 95 31.87 84
-81
---
0 95
4. Double the partial square root that you know so far, write it down, and then
append a
"mystery digit". Multiply this number by that mystery
digit, then write this product below the numbers you've written so
far. It should look like this:
9
---------------
V 81 95 31.87 84
-81
---
18_ 0 95
* _ =
5. Select the "mystery digit" so that the number is as
large as possible, but not larger than the numbers above it. In
this case the largest digit that fits is 0.
9
---------------
V 81 95 31.87 84
-81
---
180 0 95
* 0 = 0
6. Write the mystery digit, 0, at the top, then subtract, and bring
down the next pair of digits:
9
0
---------------
V 81 95 31.87 84
-81
---
180 0 95
* 0 = 0
----
95 31
Now repeat steps 4 through 6.
9
0 5
---------------
V 81 95 31.87 84
-81
---
180 0 95
* 0 = 0
----
1805 95 31
* 5 = -90 25
-----
506 87
And again, two more times. Remember to place the decimal point
in the square root just above the decimal point in the original number.
9
0 5. 2 8
---------------
V 81 95 31.87 84
-81
---
180 0 95
* 0 = 0
----
1805 95 31
* 5 = -90 25
-----
18102 506 87
* 2 = -362 04
-------
181048 14483 84
* 8 =-14483 84
---------
0
The final answer, then is:
------------
V 819531.8784 = 905.28
