Home - HomeschoolMath.net      Adapted Mind, the better way to learn math
You are here: HomeArticles → Square root algorithm example

How to use the square root algorithm to find the square root of a six-digit number

Example: Find √297 504 to two decimal places.

First group the numbers under the root in pairs from right to left, leaving either one or two digits on the left.  For each pair of numbers you will get one digit in the square root.  
To start, find a number whose square is less than or equal to the first pair (29), and write it above the square root line (5).

5
29.75.04

Next:

5
29.75.04

- 25

4

75
5
29.75.04

- 25

(10_)

4

75
5

4

29.75.04

- 25

(104)

4

75
Square the 5, giving 25, write that underneath the 29, and subtract. Bring down the next pair of digits. Then double the number above the square root symbol line (highlighted), and write it down in parenthesis with an empty line next to it as shown. Next think what single digit number something could go on the empty line so that hundred-something times something would be less than or equal to 475.
102 x 2 = 204
104 x 4 = 416, so 4 works.  Write therefore 4 in the answer.
5

4

29.75.04

- 25

(104)

4

75
- 4 16
5904
5

4

29

.75.04

- 25

(104)

4

75
- 4 16
(108_)5904
5 4 5

29

.75.04

- 25

(104)

4

75
- 4 16
(10855904
Calculate 4 x 104, write the product below 475, subtract, bring down the next pair of digits. Then double the number
in the answer (54), and write the doubled number (108) in parenthesis with an empty line next to it as shown.
Think what single digit number something could go on the empty line so that thousand-eighty-something times something would be less than or equal to 5904. 1085 x 5 = 5425, so 5 works and goes to the answer.
5 4 5

29

.75.04.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
479 00
5 4 5

29

.75.04.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(1090_)479 00
5 4 5. 4

29

.75.04.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
To continue, we need to add extra decimal zeros to our number. The steps continue in exact same manner: calculate 1085 x 5 = 5425, write that below 5904, subtract, bring down the next pair of (decimal) digits Then double the 'number' 545 which is in the answer, and write the doubled number 1090 in parenthesis with an empty line next to it. 10904 x 4 = 43616, so 4 is the next digit in the answer. To find the answer to two decimal places, we need to find the third decimal with the algorithm, so we will know whether the answer to two decimals would be rounded up or down. So two more rounds to go.
5 4 5 4

29

.75.04.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
42 84 00
5 4 5 4

29

.75.04.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
(10908_)  42 84 00
5 4 5 4 3

29

.75.04.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
(109083)  42 84 00
  
5 4 5 4 3

29

.75.04.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
(109083)  42 84 00
32 72 49

10

11 51
5 4 5 4 3

29

.75.04.00.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
(109083)  42 84 00
32 72 49
(109086_) 

10

11 51 00
5 4 5 4 3 9

29

.75.04.00.00.00

- 25

(104)

4

75
- 4 16
(1085) 5904
5425
(10904)479 00
- 436 16
(109083)  42 84 00
32 72 49
(1090869

10

11 51 00

Since the last decimal we find with the algorithm is 9, it means the previous decimal will be rounded up, and thus to one decimal place, 297504 = 545.44



See also

Should I teach my child to calculate square roots without a calculator?
Plus, learn the algorithm for calculating square roots by hand.

An explanation of why this square root algorithm works.

Free worksheets for square roots




Math Lessons menu