To find the square root of $1024$ using the division method (also known as the long division method), we will follow a systematic approach similar to long division.

Step-by-Step Solution:

1. Pair the Digits from Right to Left:
   Start by pairing the digits of the number $1024$ from right to left. We get the pairs as $(10)(24)$.

2. Find the Largest Integer:
   Find the largest integer whose square is less than or equal to the leftmost pair or single digit. In this case, the leftmost part is $10$.

   • The largest integer whose square is less than or equal to $10$ is $3$ (since $3^2 = 9$ and $4^2 = 16$ which is greater than $10$).

3. Subtract and Bring Down the Next Pair:

   • Subtract $9$ (which is $3^2$) from $10$, which gives $1$.
   • Bring down the next pair of digits (which is $24$), making the number $124$.

4. Find the Next Digit of the Root:

   • Double the current quotient (which is $3$) to get $6$.

   • Now find a digit $X$ such that $6X \times X \leq 124$.

   • The suitable $X$ is $2$ because: $62 \times 2 = 124$ This exactly matches $124$.

5. Subtract and Check for Remainder:

   • Subtract $124$ from $124$, which gives $0$.

Since there are no more pairs to bring down and the remainder is $0$, the square root calculation is complete.

Result:

The square root of $1024$ is 32.

Would you like to go through any of the steps in more detail, or do you have any questions?

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Expanded Questions:

1. How would the division method differ for a number like $1225$?
2. Can the division method be applied to find the square root of decimal numbers?
3. How do you determine the initial digit in the square root when the leftmost pair has more digits?
4. What is the best way to verify the result obtained from the division method?
5. How does the long division method for square roots differ from other methods, such as prime factorization?

Tip:

When performing the division method, keep the digits aligned and write each step clearly to avoid confusion or errors.