By Michael Avidon, math editor
You can calculate the square root of a number to any number of places without a calculator. If the original number is not a perfect square, then its square root will be irrational. Therefore, the decimal expansion is non-terminating and non-repeating. This method will give you the best approximation to any number of places.
Determine the square root of 76 to two decimal places.
First, find the greatest whole number whose square is less than 76. This is 8, as 64 < 76.
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After this, you will insert two zeros at a time after the decimal in the radicand for every decimal place in the square root. The procedure follows:
? is the greatest digit such that this product (ignoring decimals) does not exceed the remainder. Because and 7 x 167 = 1169 < 1200 and 8 x 168 = 1344 > 1200, ? = 7.
Repeat the procedure. This time, you will double 87 to get 174.
(Note that 2 x 1742 = 3484 > 3100, but the product is closer to 3100 than 1741.)
The same procedure one more time yields 8.717 (7 x 17,427 = 121,989 < 135,900). So, rounded to two decimal places, √76 = 8.72.