Find the square root of 3249 by Division method.

To find the square root of 3249 using the division method, follow these steps:

Step 1: Group the digits of the number into pairs, starting from the right. If there is an odd number of digits, the leftmost group will have only one digit.

In this case, the number is 3249, so we group it as 32 and 49.

Step 2: Find the largest number whose square is less than or equal to the leftmost group (32). The square of 5 is 25, so the quotient of the division will be 5.

Step 3: Write the quotient (5) above the rightmost group (49) and also to the right of the leftmost group (32), creating a divisor for the next step.


Step 4: Double the current quotient (5) and write it as the divisor. Also, leave some space for the remainder and the next digit of the root.


Step 5: Determine the largest possible digit to fill the space left for the remainder and the next digit of the root. This digit will be placed on top of the divisor, and its product with the divisor will be subtracted from the current dividend.

To find this digit, perform a trial multiplication by placing a digit in the tens place of the quotient and subtracting the result from the current dividend.

Place a digit (let's call it x) in the tens place of the quotient, making it 50 (5 * 10).
Multiply the divisor (10) by x+1 (50 + 1 = 51): 10 * 51 = 510.
Check if the result (510) is smaller or equal to the current dividend (3249). In this case, it is.
Subtract the result (510) from the current dividend (3249 - 510 = 2739).
Determine the value of x by finding the largest digit that can be placed in the tens place of the quotient without exceeding the result.
In this case, the largest digit that satisfies the condition is 5.


Step 6: Bring down the next pair of digits (39) next to the remainder (2739) to form a new dividend.


Step 7: Append a digit (let's call it y) to the right of the current root estimate. Divide the new dividend (2739) by twice the current root estimate (105), resulting in a new quotient digit.

Double the current root estimate (105) to get 210.
Append a digit (let's call it y) to the right of the current root estimate, making it 105y.
Divide the new dividend (2739) by 105y + y^2 (1050 + y^2).
Find the largest digit y that satisfies (1050 + y^2) * y ≤ 2739.
In this case, y is equal to 2.
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Step 8: Repeat steps 4 to 7 until all the digits have been brought down.