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To solve the problem of determining the exact perimeter of a square with an area of 524 cm², we can follow these steps:

1. **Step 1: Recall the formula for the area of a square:**
   \[
   A = s^2
   \]
   where \( A \) is the area and \( s \) is the side length of the square.

2. **Step 2: Solve for the side length:**
   Since the area is given as 524 cm², we set up the equation:
   \[
   s^2 = 524
   \]
   To find the side length, take the square root of both sides:
   \[
   s = \sqrt{524}
   \]

3. **Step 3: Simplify the square root:**
   Factor 524 into prime factors:
   \[
   524 = 4 \times 131
   \]
   Therefore:
   \[
   \sqrt{524} = \sqrt{4 \times 131} = \sqrt{4} \times \sqrt{131} = 2\sqrt{131}
   \]
   Thus, the side length of the square is \( s = 2\sqrt{131} \, \text{cm} \).

4. **Step 4: Find the perimeter:**
   The perimeter \( P \) of a square is given by:
   \[
   P = 4s
   \]
   Substituting the value of \( s \):
   \[
   P = 4 \times 2\sqrt{131} = 8\sqrt{131} \, \text{cm}
   \]

So, the exact perimeter in simplest radical form is:
\[
P = 8\sqrt{131} \, \text{cm}
\]

Would you like further explanation on any part of the solution?

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Here are 5 related questions:
1. How do you simplify square roots that involve prime factorization?
2. What is the relationship between the area and perimeter of a square?
3. How would you find the perimeter of a rectangle if the area and one side length are known?
4. Can you solve for the side length of a square if the perimeter is given instead of the area?
5. How do you calculate the diagonal of a square using its side length?

**Tip:** For problems involving squares, remember that once you know either the area or the perimeter, finding the other is straightforward using square root or multiplication techniques.

Determine the exact perimeter, in simplest radical form, of a square that has an area of 524 cm².

Learn how to calculate the perimeter of a square with an area of 524 cm² in its simplest radical form. This detailed solution uses key geometric concepts and square root simplification, perfect for students in Grades 8-10.

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