Problem 52 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{(y-6)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( |y-6| \).

Step 1: Understand the Expression

The expression given is \( \sqrt{(y-6)^2} \). This is the square root of a square, where the square is applied to the binomial \( y-6 \).

Step 2: Apply the Property of Square and Square Root

We know that for any real number \( a \), if \( a^2 \) is under a square root, \( \sqrt{a^2} = |a| \). Therefore, \( \sqrt{(y-6)^2} = |y-6| \).

Step 3: Simplify Using Absolute Value

The expression \( |y-6| \) represents the absolute value of \( y-6 \), which is equal to \( y-6 \) if \( y-6 \geq 0 \) (or \( y \geq 6 \)), and \( -(y-6) \) if \( y-6 < 0 \) (or \( y < 6 \)).

Step 4: Final Simplified Expression

The simplified expression of \( \sqrt{(y-6)^2} \) is \( |y-6| \), which depends on the value of \( y \) relative to \( 6 \).

Key Concepts

Understanding Absolute Value

The term **absolute value** describes how far a number is from zero, without considering its direction along the number line. It's like finding the distance between two points, but starting from zero.
For example, both 3 and -3 are 3 units away from zero. So, the absolute value of both numbers is 3, and we write it as \(|3| = 3\) and \(|-3| = 3\). The absolute value is always non-negative. In the context of simplifying radical expressions, knowing the absolute value is essential.
This is because when you encounter an expression like \(\sqrt{(y-6)^2}\), the absolute value helps us determine the non-negative equivalent of the expression under the square root.

• For positive or negative inputs, treat them as only positive in distance.
• Remember, it affects how expressions are interpreted mathematically.

Properties of Square Roots

Square roots are fundamental in mathematics. A square root is a value that, when multiplied by itself, gives the original number. For any non-negative real number \(a\), the square root is written as \(\sqrt{a}\).
One key property of square roots, crucial for simplifying expressions, is that the square root of a square—\(\sqrt{a^2}\)—is the absolute value of \(a\), written as \(|a|\). This property explains why \(\sqrt{(y-6)^2}\) simplifies to \(|y-6|\).
Using square root properties can help with:

• Breaking down complex expressions into simpler forms.
• Ensuring correct simplification by recognizing the absolute value.

These properties maintain mathematical accuracy and help standardize complex arithmetic into manageable equations.

Comprehending Real Numbers

Real numbers include a wide range of values that we use daily, from whole numbers and fractions to decimals and irrational numbers like \(\pi\). They are represented on a number line that extends infinitely in both directions.
The importance of real numbers in the context of solving or simplifying expressions is based on their ubiquitous nature in any mathematical problem. When working with the expression \(\sqrt{(y-6)^2}\), any real number **y** can represent a solution.
Key aspects of real numbers include:

• They encompass both rational and irrational numbers.
• The ability to perform operations, like addition, multiplication, as well as more complex operations like extracting roots.

This vast number system allows flexibility in evaluating expressions, ensuring thoroughness in solutions.