Problem 17 - Elementary Algebra

Question

Simplify each square root. $$ \sqrt{(x+9)^{2}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is x+9.

Step 1: Identify the expression within the square root

In the provided expression, we must simplify $$ \sqrt{(x+9)^{2}} $$ Here, (x+9) is the expression within the square root.

Step 2: Apply the square root property to the squared expression

Since the square root and squaring operations are inverses, applying the square root to a squared expression will simplify it and cancel out the square. So, we have: $$ \sqrt{(x+9)^{2}} = x + 9 $$ The simplified expression is x+9.

Key Concepts

Square Root Property

Understanding the square root property is essential when dealing with radical expressions in algebra. Essentially, the square root of a number or an algebraic expression that's been squared is simply the value that was originally squared. For example, if we have an expression like \(\backslash\backslash(sqrt{(x+9)^{2}}\backslash\backslash)\), applying the square root property means recognizing that squaring and square root operations are inverse of each other.

So, when you apply the square root to \((x+9)^2\), you essentially undo the squaring, leaving you with \(x+9\) as the simplified form. It's important to note that this property only applies when the values inside the radical are non-negative, to ensure that the principal, or non-negative, root is found.

Radical Expressions

What Are Radical Expressions?

Radical expressions are mathematical phrases that include a radical symbol with an index that indicates which root is being considered. The simplest form is when the index is 2, which refers to a square root. Simplifying radical expressions involves removing radicals from denominators, combining like terms, and rationalizing denominators when necessary.

For example, the expression \(\backslash\backslash(sqrt{(x+9)^{2}}\backslash\backslash)\) is a radical expression because it contains a square root. Simplifying this expression, as shown in the solution, involves recognizing that the expression inside the square root has been squared, so we can directly remove the radical by applying the square root.

Inverse Operations

Inverse operations are pairs of mathematical operations that undo each other. The most common examples are addition and subtraction, or multiplication and division. In the context of square roots, the relevant inverse operations are squaring and finding the square root. These two operations cancel each other out.

As we saw with \(\backslash\backslash(sqrt{(x+9)^{2}}\backslash\backslash)\), taking the square root of a squared expression effectively removes the square, simplifying the expression to its base, \(x+9\). It's this principle of inverse operations that allows for simplification of expressions and solving algebraic equations.

Algebraic Expressions

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can convey a variety of mathematical relationships. Simplifying an algebraic expression means performing all possible operations to get the most concise form of the expression.

In our example, \(\backslash\backslash(sqrt{(x+9)^{2}}\backslash\backslash)\), the initial expression has the appearance of complexity due to the radical. However, by understanding the underlying algebraic structure and applying the rules of inverse operations, we are able to simplify it effectively to just \(x+9\), which is an uncomplicated algebraic expression.