Find the square root of 9801 by long division method.

To find the square root of 9801 using the long division method, we can break it down into a series of logical steps. This method is quite systematic and can be applied to any number, making it a valuable technique for calculating square roots manually.

Step-by-Step Process

Let’s go through the long division method for finding the square root of 9801:

1. Pair the Digits

First, we need to pair the digits of the number starting from the right. For 9801, we can pair it as follows:

• 98 | 01

This means we have two pairs: 98 and 01.

2. Find the Largest Square

Next, we look for the largest square number that is less than or equal to the first pair (98). The largest square less than 98 is 81, which is 9 squared. So, we write:

• 9

Now, we subtract 81 from 98:

• 98 - 81 = 17

3. Bring Down the Next Pair

Now, we bring down the next pair (01) next to 17, making it 1701.

4. Double the Current Quotient

Next, we double the quotient we found (which is 9) to get 18. We will use this to find the next digit of our square root:

• 2 × 9 = 18

5. Find the Next Digit

Now we need to find a digit (let's call it 'x') such that when we append it to 18 and multiply the whole number by 'x', it is less than or equal to 1701. We can express this as:

• (180 + x) × x ≤ 1701

Testing with different digits:

• If x = 9: (180 + 9) × 9 = 189 × 9 = 1701

This works perfectly, so we write down 9 as the next digit of our square root. Now, we have:

• Square root so far: 99

6. Final Result

Since we have no more pairs to bring down, we conclude that the square root of 9801 is 99.

Verification

To ensure our result is correct, we can square 99:

• 99 × 99 = 9801

This confirms that our calculation is accurate. Therefore, the square root of 9801 is indeed 99.

Summary

The long division method is a reliable way to find square roots, especially for larger numbers. By systematically pairing digits, finding the largest squares, and carefully calculating, we can arrive at the correct answer without needing a calculator. In this case, we found that the square root of 9801 is 99.