Find the square root of 3136 by division method.

(a) 56

(b) 66

(c) 46

(d) 36

To find the square root of 3136 using the division method, let's break it down step by step. This method is particularly useful for larger numbers and involves a systematic approach to arrive at the square root.

Step-by-Step Process

1. Pair the Digits

First, we need to pair the digits of the number starting from the right. For 3136, we can pair it as (31)(36).

2. Find the Largest Square

Next, we look for the largest square number that is less than or equal to the first pair (31). The largest square less than 31 is 25, which is \(5^2\). So, we write:

• 5 (the square root of 25) goes above the line.
• 5 is then squared to give 25, which we subtract from 31, leaving us with 6.

3. Bring Down the Next Pair

Now, we bring down the next pair (36) next to the remainder (6), making it 636.

4. Double the Quotient

Next, we double the quotient we have above the line (which is 5), giving us 10. We now need to find a digit (let's call it 'x') such that:

10x * x is less than or equal to 636.

5. Determine the Digit

Testing values for 'x':

• If x = 6: 106 * 6 = 636, which fits perfectly.

So, we place 6 next to 5 above the line, making it 56.

6. Final Calculation

Now, we subtract 636 from 636, leaving us with 0. Since there are no more pairs to bring down, we conclude that:

The square root of 3136 is 56.

Conclusion

Thus, the answer to your question is (a) 56. This method not only helps in finding the square root but also enhances your understanding of numbers and their properties.