Finding the square root of a number using long division can be a bit tedious, but it can be done. Here's how you can find the square root of 1015 using the long division method:
Step 1: Group the digits in pairs from right to left, starting with the decimal point (if there's no decimal point, assume it's at the end):
1,015
Step 2: Find the largest digit "x" such that when you square it, it's less than or equal to the leftmost group (1 in this case). In this case, the largest digit that works is 3 because 3^2 = 9, which is less than 10 (the leftmost group). So, the first digit of the square root is 3.
Step 3: Write down the 3 as the first digit of the square root and subtract the square of this digit (9) from the leftmost group:
3
______
1,015
- 9
-----
Now, bring down the next pair of digits (15) to the right of the result, and double the first digit of the current result (which is 3) to form a "dividend" for the next step. We want to find a digit "x" to place to the right of the 3 to complete a new "dividend" that is equal to or less than 1515.
Step 4: Find a digit "x" to place to the right of the 3, such that when the new dividend is formed (33x), the product of this new dividend and x is less than or equal to 1515. You can use trial and error to find this digit. In this case, the largest digit that works is 5 because 335 times 5 is 1675, which is greater than 1515. So, place 4 (the next smallest digit) to the right of the 3:
34
______
1,015
- 9
-----
Step 5: Now, find the product of the new dividend (344) and the last digit (4) and write it below the 1515:
34
______
1,015
- 9
-----
136
Step 6: Bring down the next pair of digits (00) to the right of the result, and double the current result (344) to form a new dividend for the next step.
Step 7: Find a digit "x" to place to the right of 344, such that when the new dividend (344x) is multiplied by x, it is less than or equal to 13600. Again, you can use trial and error. In this case, the largest digit that works is 3 because 3443 times 3 is 10329, which is less than 13600.
3443
______
1,015
- 9
-----
13600
Step 8: Repeat steps 5, 6, and 7 until you have the desired level of accuracy. You can continue with more decimal places if needed.
Continue this process until you have as many decimal places as you desire. The square root of 1015 is approximately 31.8590646441.