Find the square root of 1024 by division method?

(a) 42

(b) 48

(c) 38

(d) 32

To find the square root of 1024 using the division method, we can break down the process step by step. This method is systematic and helps in understanding how square roots can be calculated without relying solely on a calculator.

Step-by-Step Process

1. Set Up the Problem

First, we need to pair the digits of the number from right to left. For 1024, we pair it as (10)(24). This means we will treat '10' as one pair and '24' as another.

2. Find the Largest Square

Next, we look for the largest square number that is less than or equal to the first pair (10). The largest square is 3, since \(3^2 = 9\). We write 3 above the pair and subtract \(9\) from \(10\):

• 3 (the quotient)
• 10 - 9 = 1 (the remainder)

3. Bring Down the Next Pair

Now, we bring down the next pair (24) next to the remainder (1), making it 124.

4. Double the Quotient

We then double the quotient we found (which is 3) to get 6. This will be the starting point for our next divisor.

5. Find the Next Digit

Now, we need to find a digit (let's call it 'x') such that when we form the number (60 + x) and multiply it by 'x', the result is less than or equal to 124. We can test digits:

• If x = 2: (60 + 2) * 2 = 62 * 2 = 124 (this works perfectly)

So, we place '2' next to the '3' in the quotient, making it 32.

6. Final Calculation

Now, we subtract \(124\) from \(124\) to get \(0\). Since there are no more pairs to bring down, we have completed the process.

Result

The square root of 1024 is 32. Therefore, the correct answer is (d) 32.

Verification

To verify, we can square 32:

• 32 * 32 = 1024

This confirms that our calculation is correct. The division method is a reliable way to find square roots, especially for larger numbers, and it helps reinforce our understanding of multiplication and division.