To find the square root of a number by the division method (also called the long division method), follow these steps:
Steps to Find Square Root by Division Method
- Pair the digits of the number starting from the decimal point (or unit place) towards left and right, grouping in pairs of two digits each.
- Find the largest number whose square is less than or equal to the leftmost pair (or single digit if odd number of digits). This number is the first digit of the square root (quotient).
- Subtract the square of this number from the leftmost pair and bring down the next pair of digits next to the remainder to form a new dividend.
- Double the quotient obtained so far and write it as the new divisor's first part, leaving a blank space for the next digit.
- Find the largest digit to fill the blank in the divisor such that when the new divisor is multiplied by this digit, the product is less than or equal to the current dividend.
- Write this digit as the next digit of the quotient.
- Subtract the product from the dividend to get the new remainder.
- Bring down the next pair of digits and repeat steps 4 to 7 until all pairs are used or desired decimal accuracy is reached.
- If the number is not a perfect square, you can add pairs of zeros after the decimal point and continue the process to get decimal places in the square root.
Example: Find √484 by division method
- Pair digits: 4 | 84
- Largest square ≤ 4 is 2 (since 2²=4). Quotient = 2, remainder = 0
- Bring down 84 → new dividend = 84
- Double quotient: 2 × 2 = 4 → divisor starts with 4_
- Find digit to fill blank: 42 × 2 = 84 ≤ 84
- Quotient becomes 22, remainder 0
- No more digits left, so √484 = 22
This method works for both perfect squares and non-perfect squares (with decimals) and is a manual way to find square roots without a calculator
. You can continue the process for decimal places by adding pairs of zeros after the decimal point in the dividend and repeating the steps. This method is systematic and involves repeated divide, multiply, subtract, bring down, and repeat steps until the desired precision is achieved
