In Exercises 33–46, simplify each expression._____√(x−1)²

Verified step by step guidance

1. Recognize that the expression under the square root is \((x-1)^2\).
2. Recall the property of square roots: \(\sqrt{a^2} = |a|\), where \(|a|\) denotes the absolute value of \(a\).
3. Apply this property to the expression: \(\sqrt{(x-1)^2} = |x-1|\).
4. Understand that the absolute value \(|x-1|\) represents the distance of \(x-1\) from zero on the number line, ensuring the result is non-negative.
5. Conclude that the simplified form of the expression is \(|x-1|\).

Key Concepts

Square Root Function

The square root function, denoted as √x, is the inverse of squaring a number. It returns the non-negative value that, when squared, gives the original number. For example, √(x²) = |x|, which means the square root of a squared term is the absolute value of that term.

Absolute Value

Absolute value, represented as |x|, measures the distance of a number from zero on the number line, regardless of direction. It is always non-negative. In the context of square roots, when simplifying expressions like √(x−1)², the result is |x−1|, indicating that the output will be the positive distance from x to 1.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form while maintaining its value. This often includes combining like terms, factoring, and applying properties of operations. In this case, simplifying √(x−1)² requires recognizing that it can be expressed as |x−1|, which is a more concise representation of the original expression.