Answer:
The square root of 18769 is exactly 137.
Explanation:
To find the square root of a number using the division method (also known as the long division method), follow these steps:
Group the digits of the number in pairs from right to left. If the number of digits is odd, the leftmost group will have only one digit.
Find the largest digit "x" such that when "x" is squared and placed to the left of the first pair, the result is less than or equal to the first pair. Write "x" above the line.
Subtract the product of "x" squared and the first pair from the first pair, and bring down the next pair of digits. This gives you a new dividend.
Double the value of "x" and put a placeholder (0) to its right. This becomes your divisor.
Perform long division: Divide the new dividend by the new divisor, and write the result above the line.
Multiply the complete divisor (including the placeholder) by the result, and write the product below the new dividend. Subtract this product from the new dividend to get a new remainder.
Bring down the next pair of digits from the original number to the right of the new remainder.
Repeat steps 2 to 7 until all pairs of digits have been used.
Let's find the square root of 18769 using the division method:
Group the digits: 18769 → 1 | 87 | 69
Find the largest "x" such that x^2 ≤ 1. The largest such x is 1.
Subtract: 1 - 1^2 = 0. Bring down 87.
Double x and add a placeholder: 1 * 2 = 2, divisor = 20.
Divide: 87 ÷ 20 = 4 (write above the line).
Multiply: 20 * 4 = 80 (write below the line).
Subtract: 87 - 80 = 7. Bring down 69.
New dividend: 769.
Find the largest "x" such that x^2 ≤ 769. The largest such x is 8.
Subtract: 769 - 8^2 * 100 = 169.
Since we've exhausted all the pairs, our final answer is the combination of the quotients we found in step 5: 14.
Therefore, the square root of 18769 is 137 using the division method.