The document outlines a method for calculating the square root of 8259876 without a calculator by using geometric interpretation and a guess and check approach. It systematically identifies the values of the thousands, hundreds, tens, and ones digits needed to arrive at the square root, ultimately determining it to be 2874. The accuracy of the calculation is verified by multiplying the found root to confirm it equals the original number.

2. How can you find the square root of 8259876 without a calculator? Submitted for Mr. Stafford’s Class, 2nd Period Algebra 1 Name: John Doe Period: 3rd

3. First, think geometrically. A square number is just a square with side lengths of that number

4. And you can split this square into smaller squares and rectangles a = 1000’s value of square root b = 100’s value of square root c = 10’s value of square root d = 1’s value of square root SO a + b + c + d = Square Root

5. Each of these rectangles has it’s own area, which when added together A = a 2 + 2(a x b) + b 2 + 2(c x (a + b)) + c 2 + 2 (d x (a + b + c))+ d 2

6. a =? b =? c =? d = ? The area of this full box is, and to find the square root of 8259876 we’re going to use the geometry to methodically find the areas of each of it’s smaller pieces using a guess and check method. We’ll start with the largest square, a 2 , and try to find the thousands value Procedure - Guess the 1000’s digit squared until it’s bigger than the number we used, then go lower. Guess 1 - 4000 2 = 16,000,000 too big! Guess 2 - 3000 2 = 9,000,000 too big! Guess 3 - 2000 2 = 4,000,000 this works! Now, subtract this number from the original 8259876 - 4,000,000 = 4,259876 A = a 2 + 2(a x b) + b 2 + 2(c x (a + b)) + c 2 + 2 (d x (a + b + c))+ d 2

7. a =2000 b =? c =? d = ? Now we’ll try finding the hundreds value, and work with the next three largest boxes Procedure - Guess the 100’s digit that when squared and multiplied by 2a is less than 4,259,876 Guess 1 - 400 2 + 2(2000 x 400 )= 1,760,000 too small! Guess 2 - 900 2 + 2(2000 x 900 ) = 4,410,000 too big! Guess 3 - 800 2 + 2(2000 x 800 ) = 3,840,000 this works! Now, subtract this number from the remainder 4,259,876 - 3,840,000 = 419,876 A = a 2 + 2(a x b) + b 2 + 2(c x (a + b)) + c 2 + 2 (d x (a + b + c))+ d 2

8. a =2000 b =800 c =? d = ? Now we’ll try finding the tens value, and work with the next three largest boxes Procedure - Guess the 10’s value that when squared and multiplied by 2(a+b) is less than 419,876 Guess 1 - 50 2 + 2( 50 x (2000 + 800))= 282,500 too small! Guess 2 - 70 2 + 2( 70 x (2000 + 800))= 396,900 too small! Guess 3 - 80 2 + 2( 80 x (2000 + 800))= 454,400 too big! So the number that works is 70 ! Now, subtract this number from the remainder 419,876 - 396,900 = 22,976 A = a 2 + 2(a x b) + b 2 + 2(c x (a + b)) + c 2 + 2 (d x (a + b + c))+ d 2

9. a =2000 b =800 c =70 d = ? Now we’ll try finding the ones value, and work with the next three largest boxes Procedure - Guess the 1’s value that when squared and multiplied by 2(a+b +c) is less than 419,876 Guess 1 - 5 2 + 2( 5 x (2000 + 800 + 70))= 28,725 too big! Guess 2 - 4 2 + 2( 4 x (2000 + 800 + 70))= 22,976 This works perfectly! Subtract from the remainder 22,976 - 22,976 = 0! A = a 2 + 2(a x b) + b 2 + 2(c x (a + b)) + c 2 + 2 (d x (a + b + c))+ d 2

10. So we think our number is 2000 + 800 + 70 + 4 = 2874. We can check our work to see if we’ve found the correct root! 2874 x2874 11496 201180 2299200 5748000 8259876 8259876 1/2 = 2874 This was our original number, so we’ve done it correctly!

11. Thanks! - John Doe