Problem 39 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{4 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2|x|\).

Step 1: Recognize the Expression

We are given the expression \( \sqrt{4x^2} \). This is the square root of a perfect square expression, where 4 can be rewritten as \(2^2\).

Step 2: Apply the Square Root Property

The square root of a square \(\sqrt{a^2}\) is \(|a|\). Therefore, we rewrite the expression as \(\sqrt{(2x)^2}\), which simplifies to \(|2x|\).

Step 3: Simplify Using Absolute Values

Since \(|2x| = 2|x|\), we simplify the expression to \(2|x|\). This is because absolute value ensures non-negativity, which is critical when dealing with square roots.

Key Concepts

Square Roots

Square roots are fundamental in mathematics and are denoted by the radical symbol \( \sqrt{} \). To understand square roots, we need to identify the number or expression inside the root, known as the radicand. The principal square root of a positive number \( a \) is the number \( b \) such that \( b^2 = a \).
For instance, the square root of 9, written as \( \sqrt{9} \), is 3 because \( 3^2 = 9 \).When dealing with expressions like \( \sqrt{4x^2} \), it is helpful to first identify whether the radicand is a perfect square. If so, you simplify by finding the square of each component. For example:- Recognize 4 as \( 2^2 \).- Recognize \( x^2 \) as a square.So \( \sqrt{4x^2} \) simplifies essentially by applying the property \( \sqrt{a^2} = |a| \), resulting in \( |2x| \). This approach is particularly useful in algebra where simplifying expressions leads to more manageable forms.

Absolute Values in Algebra

Absolute values are important when simplifying algebraic expressions. They denote the distance from zero on a number line, which is always non-negative. In algebra, the absolute value of a number \( a \) is written as \( |a| \) and respects the properties:

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

When solving equations or simplifying expressions like \(|2x|\), the role of the absolute value is to ensure the result is not negative, regardless of the variable involved. In our expression \( \sqrt{4x^2} = |2x| = 2|x| \): - The absolute value represents positive distance ensuring correctness in contexts where \( x \) may vary, such as real numbers. Consider this function-like characteristic as a safeguard when variables can take any real number value.

Perfect Squares in Algebra

Understanding perfect squares is crucial for simplifying square root expressions. A perfect square is any number or expression that can be expressed as the square of another integer or algebraic expression. For instance, numbers like 4, 9, or 16 are perfect squares because they can be written as \(2^2\), \(3^2\), and \(4^2\), respectively.In algebra: - Expressions such as \(x^2\), \((3x)^2\), are perfect squares where each component inside the square is squared. Recognizing perfect squares simplifies processes involving square roots like in the given example \( \sqrt{4x^2} \). By saying that the radicand \(4x^2\) is a perfect square, we can use the property \( \sqrt{a^2} = |a| \) directly. This transforms a potentially complex algebraic problem into a simpler task, allowing one to reframe it as \(|2x|\), then \(2|x|\), keeping mathematical integrity while navigating through algebraic manipulations. Understanding these principles systematically reduces complexity.