If at least how many soldiers are removed from 56728 soldiers or at least how many more soldiers are added to them, the army can be arranged in a square?

To determine how many soldiers should be removed or added to make the total number of soldiers form a perfect square, we need to find the nearest perfect square to 56,728.

A perfect square is any number that is the square of an integer. We will:

1. Find the square root of 56,728.
2. Round the square root to the nearest integers.
3. Calculate how many soldiers need to be added or removed to reach the nearest perfect squares.

### Step 1: Find the square root of 56,728
\[
\sqrt{56728} \approx 238.13
\]
This means the nearest integers to the square root are 238 and 239.

### Step 2: Find the nearest perfect squares
- \( 238^2 = 238 \times 238 = 56644 \)
- \( 239^2 = 239 \times 239 = 57121 \)

### Step 3: Calculate the differences
- To make the army a perfect square by **removing soldiers**: 
\[
56728 - 56644 = 84 \text{ soldiers}
\]
So, at least **84 soldiers** need to be removed.

- To make the army a perfect square by **adding soldiers**:
\[
57121 - 56728 = 393 \text{ soldiers}
\]
So, at least **393 soldiers** need to be added.

### Final Answer:
- At least **84 soldiers** should be removed, or
- At least **393 soldiers** should be added to make the army a perfect square.

Would you like more details, or do you have any other questions?

Here are some related questions:
1. How do you calculate a perfect square?
2. What is the significance of perfect squares in real-life applications?
3. How can square roots be approximated without a calculator?
4. What is the difference between perfect squares and other types of numbers?
5. How can I find the next perfect square after a certain number?

**Tip:** When finding square roots, always check the nearest whole numbers, as they will lead to the closest perfect squares.

Discover how to calculate the number of soldiers to add or remove from 56728 to make the army a perfect square. This problem explores square roots and perfect squares, providing detailed solutions for how to arrange the army in a square formation. Suitable for students in Grades 8-10.

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