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The perimeter of a square is the total length around the outside of the square. It's simply the sum of the lengths of all four sides. Understanding the formula for finding the perimeter is critical:

• The formula is: \( P = 4s \), where \( s \) represents the length of one side.

Given this straightforward formula, calculating the perimeter involves nothing more than multiplying the length of one side by four. For example, if a square has side lengths of \( 2 \, m \), its perimeter is calculated as \( 4 \times 2 = 8 \, m \).
In our exercise, the side of the square isn't a simple number but rather a rational expression: \( \frac{5}{x-2} \). This demonstrates that the concept of perimeter isn't limited to simple numbers but can include complex expressions, making the understanding of rational expressions important.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value or set of values. These are foundational in algebra, allowing us to express relationships and solve equations.

• A simple example: \( 3x + 2 \) is an expression where "\( 3x \)" means three times a variable \( x \), plus two.

In the exercise, we encountered an expression for the side of the square: \( \frac{5}{x-2} \).
This is a bit more complex, as it introduces a rational expression, which involves a fraction. Here, the numerator is 5, and the denominator is \( x-2 \).
Understanding each part of an algebraic expression helps in substituting them into formulas like the perimeter formula, making calculations manageable. Working with algebraic expressions often involves simplifying or manipulating them to solve problems or express ideas clearly.

Simplifying expressions is a crucial step in ensuring that our mathematical expressions are as straightforward as possible while retaining their values. This step is essential for solving equations and understanding solutions.

• To simplify the perimeter expression for our square: we substitute the side length \( \frac{5}{x-2} \) into the perimeter formula.

The perimeter thus becomes: \( P = 4 \times \frac{5}{x-2} \). To simplify:

• We multiply the numerator (4 times 5), resulting in 20.
• So, the numerator of the simplification is 20 and the denominator remains \( x-2 \).

The simplified expression for the perimeter is: \( \frac{20}{x-2} \).
This process of simplifying helps to make the expression easier to interpret and use, especially when dealing with equations or further calculations. It also highlights the importance of understanding operations on rational expressions, a key skill in algebra.