Square Root of 397
2026-02-28 13:51 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-397

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 397, we can use 3 | 97.

Step 2: Now, we need to find a number whose square is less than or equal to 3. The number is 1 because 1 × 1 = 1. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 97, making the new dividend 297. Double the current quotient and add a digit to form a new divisor. The current quotient is 1, so 2 × 1 = 2.

Step 4: Now find a digit, n, such that 2n × n is less than or equal to 297. Let's consider n as 9, so 29 × 9 = 261.

Step 5: Subtract 261 from 297, which leaves 36. The quotient is now 19.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to continue with the process.

Step 7: Add two zeroes to the dividend, making it 3600.

Step 8: Now find a new divisor. 2 × 19 = 38, and find n such that 38n × n ≤ 3600. Let's consider n as 9, so 389 × 9 = 3501.

Step 9: Subtract 3501 from 3600, which leaves 99.

Step 10: Now the quotient is 19.9.

Step 11: Continue doing these steps until we reach the desired decimal precision.

So the square root of √397 is approximately 19.933.