Find the square root of the following by long division method

Find the square root of the following by long division method. (ii) 385641

The correct answer and explanation is:

Square Root of 385641 by Long Division Method

Answer: √385641 = 621

Step-by-Step Long Division Method:

1. Group the digits in pairs from right to left:
   385641 → (3)(85)(64)(1) → but it’s more practical to group as (385)(641).
2. Find the largest number whose square is less than or equal to 385:
   19² = 361, and 20² = 400 (too big), so start with 19.
   Write 19 as the first digit of the root. Subtract 361 from 385:
   385 – 361 = 24. Bring down the next pair (64), making it 2464.
3. Double the root so far (19 × 2 = 38).
   Now, we need to find a digit X such that:
   (380 + X) × X ≤ 2464. Try values:
   • 381 × 1 = 381
   • 382 × 2 = 764
   • 383 × 3 = 1149
   • 384 × 4 = 1536
   • 385 × 5 = 1925
   • 386 × 6 = 2316
   • 387 × 7 = 2709 (too big)
   So, take 6.
   (380 + 6) × 6 = 386 × 6 = 2316
   Subtract: 2464 – 2316 = 148. Bring down the next pair (41), making it 14841.
4. Double the new root (196 × 2 = 392).
   Now find X such that (3920 + X) × X ≤ 14841:
   • 3921 × 1 = 3921
   • 3922 × 2 = 7844
   • 3923 × 3 = 11769
   • 3924 × 4 = 15696 (too big)
   So, take 3.
   (3920 + 3) × 3 = 3923 × 3 = 11769
   Subtract: 14841 – 11769 = 3072

Now, 621 is the square root of 385641. Since remainder 3072 is left and all digits are used, and we only want the whole number square root, the answer is 621.

Explanation (300 Words):

The long division method is a systematic way of finding the square root of large numbers. It works by grouping the digits in pairs from right to left for whole numbers. This helps identify parts of the number whose square roots can be calculated step-by-step.

To begin, digits are grouped so that each part becomes manageable. The process starts by estimating the square root of the first group. For 385641, we group it as 385 and 641. The square root of 385 is approximately 19 because 19 squared is 361, which is closest without going over. Subtracting this leaves 24. Next, the next pair of digits, 64, is brought down to make 2464.

The next step doubles the root found so far (19 becomes 38) and determines the next digit by trial and error, ensuring the multiplication product remains below the number. This step adds another digit to the root. Repeating this with the next pair of digits eventually gives the complete root. In this case, the next digit is 6, giving 196. Doubling 196 gives 392, and the final digit 3 is chosen to keep the result below 14841. The resulting square root is 621.

This method gives an accurate whole number square root. The steps involve estimating, trial and error, multiplication, and subtraction. It is useful when calculators are not allowed or when one wants to understand how square roots are determined manually.