Problem 66 - Algebra 2

Question

Simplify. \(\sqrt{(y+2)^{2}}\)

Step-by-Step Solution

Answer

The simplified form is \(|y+2|\).

Step 1: Understanding the Square and Square Root Relationship

The expression involves simplifying a square root that encompasses a square. Specifically, we are looking at \(\sqrt{(y+2)^2}\). This is a classic case where the square and the square root are inverses of each other, meaning they "cancel each other out."

Step 2: Applying the Concept of Absolute Value

Though squaring and square rooting a term cancels out, we must remember that the square of a number results in a non-negative number. Thus, when simplifying, we need to consider the absolute value. Therefore, \(\sqrt{(y+2)^2} = |y+2|\).

Step 3: Expressing the Final Simplified Form

The simplified form of the expression \(\sqrt{(y+2)^2}\) is \(|y+2|\) because the square root of a square yields the absolute value of the base.

Key Concepts

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering direction. Hence, absolute value is always a non-negative number. For example, the absolute value of both \(3\) and \(-3\) is \(3\).
When simplifying square roots that encompass a square, like \(\sqrt{(y+2)^{2}}\), we introduce absolute value. This is because squaring any real number results in a positive number, or zero. Thus, the initial equation \(\sqrt{(y+2)^{2}}\) resolves to \(|y+2|\).
This absolute value ensures that the expression remains consistent and correct, maintaining a non-negative outcome of \(y+2\), regardless of whether \(y\) itself is positive or negative.

Inverse Operations

Inverse operations refer to pairs of mathematical operations that reverse the effect of each other. Common examples include addition with subtraction and multiplication with division.
Squaring and square rooting are also inverse operations. When we square a number, we find the product of a number with itself. Simultaneously, the square root of a number identifies the original number that was squared.
In the exercise \(\sqrt{(y+2)^{2}}\), the operations inside the square root, specifically the squaring of \(y+2\), are reversed by the square root. This inverse relationship allows us to simplify the expression directly to the absolute value \( |y+2| \).
Understanding inverse operations helps us recognize how different mathematical functions interact and cancel out, particularly in expressions involving squares and square roots.

Simplification

Simplification is the process of reducing a complex expression to its simplest form, making it easier to manage and understand. In mathematics, this often involves minimizing operations and removing parentheses where possible.
In the given problem, simplifying involves applying the relationship between square roots and squares, as well as recognizing the need for absolute value.
The expression \(\sqrt{(y+2)^{2}}\) becomes \( |y+2| \) through simplification, resulting in a cleaner, more straightforward format. The absolute value notation reflects the fact that no matter the value of \(y\), \(y+2\) maintains a non-negative state in this expression.
By simplifying expressions, we aim to make calculations easier while preserving the original values, which in this case, means remembering to apply absolute value.