Problem 51 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{(x-5)^{2}} $$

Step-by-Step Solution

Answer

The simplification of \( \sqrt{(x-5)^2} \) is \( |x-5| \).

Step 1: Understand the Square Root

The exercise asks us to simplify the expression \( \sqrt{(x-5)^2} \). The square root function \( \sqrt{\cdot} \) will "undo" the squaring process of \((x-5)\), leading to an expression that "undoes" the square function inside the root.

Step 2: Simplify Using Absolute Value

The expression \( \sqrt{(x-5)^2} \) simplifies to \( |x-5| \), which is the absolute value of \( x-5 \). The absolute value operation ensures that the expression inside is non-negative, which is consistent with the definition of a square root, which only returns the non-negative root.

Step 3: Final Simplified Expression

The final simplified result of \( \sqrt{(x-5)^2} \) is \( |x-5| \). This means that no matter whether \( x-5 \) is positive or negative, the simplified expression will always be non-negative.

Key Concepts

Simplifying Square Roots

Square roots are a fundamental mathematical concept often encountered when simplifying expressions and solving equations. The square root of a number \( n \), expressed as \( \sqrt{n} \), is a value that, when multiplied by itself, equals \( n \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). The square root symbol \( \sqrt{} \) indicates this operation.
In terms of expressions, like the one provided \( \sqrt{(x-5)^2} \), the square root effectively "cancels out" the squaring of \((x-5)\). However, this cancellation is only straightforward when dealing with non-negative numbers as square roots are designed to provide the principal (non-negative) root.To simplify square roots:

• Identify the expression under the square root.
• If it is a perfect square like \((x-5)^2\), you can simplify it using the absolute value function.

Understanding Absolute Value

Absolute value measures the distance of a number from zero on a number line, regardless of direction, thus ensuring a non-negative output. Denoted by \(|x|\), the absolute value of \(x\) is defined as follows:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

The need for absolute value arises whenever we deal with square roots of squared variables, like \( \sqrt{(x-5)^2} \). As squaring a number always results in a non-negative value, the square root must also ensure a non-negative result. Thus, \( \sqrt{(x-5)^2} = |x-5| \).
This result aligns with the property of absolute value, as it guarantees that the expression inside the square root remains non-negative, fulfilling the requirement that square roots yield non-negative outcomes.

The Role of Real Numbers in Simplification

Real numbers encompass all the numbers on the number line, including both rational numbers (like integers and fractions) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)). When simplifying expressions, the assumption is often made that variables, such as \(x\), can represent any real number, which influences how we approach the simplification.
In the context of simplifying an expression like \( \sqrt{(x-5)^2} \), saying that \(x\) can be any real number highlights that the value under the square root can be positive, negative, or zero. Thus, applying absolute value ensures the result remains non-negative no matter the real number value of \(x\). This understanding is crucial because while some constraints apply in different contexts (for example, avoid dividing by zero), here, we focus on ensuring that operations like taking a square root are valid for any real number input.