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To simplify square roots, the goal is to find the prime factors of the number within the square root and determine if any of those factors are perfect squares. A perfect square is an integer that is the product of an integer multiplied by itself. For example, 16 is a perfect square because it can be written as 4 times 4.

When simplifying the square root of an algebraic expression, look for factors that are perfect square trinomials, such as the one in our example \(x^2 - 10x + 25\)). Recognizing that this trinomial is the same as \(x - 5)^2\)) allows you to replace the original square root expression with \(\sqrt{(x - 5)^2}\)) which then simplifies to \(x - 5\)). However, since the square root is also an even function, meaning that \(\sqrt{x^2} = |x|\)), we consider the absolute value. Therefore, \(\sqrt{(x - 5)^2} = |x - 5|\)), making sure our simplified form accounts for both the positive and negative roots.

When it comes to factoring trinomials, particularly perfect square trinomials, you look for expressions of the form \(x^2 + 2ab + b^2\)) or \(x^2 - 2ab + b^2\)). These trinomials factor into \( (x + b)^2 \)) and \( (x - b)^2 \)) respectively. In our exercise, \(x^2 - 10x + 25\)) is a prime example of such a trinomial that factors neatly into \( (x - 5)^2 \)).

The process involves identifying the square root of the first and last term, then determining if twice the product of these roots give the middle term. Here, since the square root of \(x^2\)) is \(x\)) and the square root of 25 is 5, and indeed, twice 5 times \(x\)) yields 10x (the middle term with its sign), we confirm that this is a perfect square trinomial that factors as demonstrated.

Understanding mathematical expressions is critical for simplifying and manipulating algebraic equations. An expression can include variables, constants, and operational symbols that, when evaluated, result in a value. Expressions can be classified into types such as monomials, polynomials, and trinomials—the latter involving three terms.

In our scenario, the expression \( -\sqrt{x^{2}-10 x+25} \)) is a combination of operations and an algebraic trinomial within a square root. Simplification of such an expression involves recognizing underlying perfect squares and executing algebraic operations while adhering to the rules of square roots and absolute values. The simplification process enhances the understanding of the structure of mathematical expressions and the intricate relationships between their components.