To find the square root of 280567 using the long division method, follow these steps:

Step 1: Group the Digits

Starting from the unit place, group the digits of the number in pairs (if there is an odd digit, it stands alone). For 280567, we can group it as:

28 | 05 | 67

Step 2: Find the Largest Square

Find the largest square number less than or equal to the first group (28 in this case).

• $5^2 = 25$, and $6^2 = 36$ (which is greater than 28), so we take $5^2 = 25$.

Step 3: Subtract and Bring Down the Next Group

• Subtract 25 from 28, leaving a remainder of 3.
• Bring down the next pair (05), making it 305.

Step 4: Double the Quotient and Find the Next Digit

• Double the current quotient (which is 5), giving us 10.
• We now need to find a digit $x$ such that $10x \times x$ is less than or equal to 305.
• $102 \times 2 = 204$
• $103 \times 3 = 309$ (which is greater than 305), so $x = 2$.

Step 5: Subtract and Bring Down the Next Group

• Subtract $204$ from $305$, leaving a remainder of $101$.
• Bring down the next pair (67), making it $10167$.

Step 6: Repeat the Process

• Double the quotient (which is now 52), giving us 104.
• We need to find a digit $y$ such that $104y \times y$ is less than or equal to $10167$.
• $1049 \times 9 = 9441$
• $1050 \times 0 = 10500$ (which is greater than 10167), so $y = 9$.

Step 7: Subtract and Final Result

• Subtract $9441$ from $10167$, leaving a remainder of $726$.

The quotient (the digits found) is 529, which means: $\sqrt{280567} \approx 529$

The remainder (726) suggests that 280567 is not a perfect square, so the actual square root is slightly more than 529, but the result 529 is the whole number closest to the square root using the long division method.

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Would you like more details or have any questions? Here are 8 related questions to deepen your understanding:

1. How do you find square roots of decimal numbers using the long division method?
2. What is the difference between exact square roots and approximate square roots?
3. Can you explain how to find cube roots using a similar method?
4. Why is the long division method useful for finding square roots?
5. How can you estimate square roots without a calculator?
6. What are some alternative methods for finding square roots?
7. How do you check if a number is a perfect square?
8. Can this method be applied to finding roots of larger numbers?

Tip: Understanding the process behind the long division method for square roots can improve your number sense and help in approximating roots mentally.