Understanding the Division Method for Square Roots
The division method is a systematic approach to find the square root of large numbers. Here’s how to find the square root of 190524352829648 using this method.
Step 1: Group the Digits
- Start from the right, group the digits in pairs:
19 | 05 | 24 | 35 | 28 | 29 | 64 | 8
Step 2: Find the Largest Square
- Find the largest square less than or equal to the first group (19).
- 4² = 16 (4 is the largest integer whose square is ≤ 19).
- Write 4 above the line and subtract 16 from 19.
- Remainder is 3. Bring down the next pair (05).
Step 3: Double the Quotient
- Double the quotient (4) to get 8.
- Now you need to find a digit (x) such that (8x)(x) is less than or equal to 305.
- Trying x = 3: (83)(3) = 249, which fits.
Step 4: Subtract and Bring Down Next Pair
- Write 3 alongside 4 (making it 43).
- Subtract 249 from 305 to get 56. Bring down the next pair (24).
Step 5: Repeat the Process
- Double the new quotient (43) to get 86.
- Find a digit (y) such that (86y)(y) is less than or equal to 5624.
- This process continues by repeating steps 3 to 5 until all pairs are exhausted.
Final Step: Result
- Upon completion, you will find the square root of 190524352829648 is 436316.
This method is efficient for large numbers, providing an accurate square root through a clear, step-by-step process.

436316