Find the sq root of 10 by Division method. 

To find the square root of 10 using the long division method, we follow a step-by-step process. This method is a traditional approach to calculating square roots manually, and it can be extended to find a decimal approximation of the square root of a non-perfect square like 10. 

### Steps:

1. **Pair the digits of the number starting from the decimal point**:
   - In this case, the number is 10. So we write it as \( 10.000000 \) (we can extend it with as many zeros as needed for the desired precision).

2. **Find the largest number whose square is less than or equal to the first pair**:
   - The first pair is 10. The largest integer whose square is less than or equal to 10 is 3 because \( 3^2 = 9 \). So, the quotient is 3, and the divisor is 3. Write 3 above the division line.
   
3. **Subtract the square of 3 from the first pair**:
   - \( 10 - 9 = 1 \). Bring down two zeros to make the number 100.

4. **Double the quotient for the new divisor**:
   - Double 3 to get 6. Now find a digit (let's call it X) such that \( (60 + X) \times X \) is less than or equal to 100. The largest such X is 1 because \( (60 + 1) \times 1 = 61 \).

5. **Subtract and bring down the next pair of zeros**:
   - \( 100 - 61 = 39 \). Bring down two more zeros, making it 3900.

6. **Update the divisor and repeat**:
   - Now, double the part of the quotient we have, which is 31, to get 62. We need to find the largest X such that \( (620 + X) \times X \) is less than or equal to 3900. The largest such X is 6 because \( (620 + 6) \times 6 = 3726 \).

7. **Subtract and bring down the next pair of zeros**:
   - \( 3900 - 3726 = 174 \). Bring down two more zeros, making it 17400. Continue the process to get more decimal places.

After continuing this process, we find that:

\[
\sqrt{10} \approx 3.162
\]

Thus, the square root of 10, up to three decimal places, is approximately **3.162**.

Would you like more details on any part of this process?

### Related Questions:
1. How do you find square roots for numbers larger than 100 using the division method?
2. What is the square root of 5 using the division method?
3. How does the division method compare with other methods of finding square roots, such as Newton’s method?
4. Can the square root of a fraction (e.g., 1/2) be found using the long division method?
5. How do we extend the division method for roots other than square roots, like cube roots?

### Tip:
Practice the division method with smaller numbers first to become familiar with the process before moving on to larger or more complex numbers.

Find the sq root of 10 by Division method.

Learn how to calculate the square root of 10 using the long division method. This step-by-step guide demonstrates how to manually approximate the square root of non-perfect squares, providing a solution accurate to three decimal places. Suitable for Grades 8-10.

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