Problem 53 - Algebra A Combined Function

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{4 x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{4x^2} \) is \( 2|x| \).

Step 1: Understand the Problem Statement

The problem asks for the simplification of the square root expression \( \sqrt{4x^2} \). To do this, we need to find an equivalent expression that is simpler or in a more straightforward form.

Step 2: Identify the Square Root Properties

Recall the property of square roots: \( \sqrt{a^2} = a \) provided \( a \geq 0 \). Also, if \( a < 0 \), then \( \sqrt{a^2} = |a| \). This property will help us break down and simplify the expression.

Step 3: Break Down the Expression

The expression \( \sqrt{4x^2} \) can be rewritten using the square root multiplication property as \( \sqrt{4} \cdot \sqrt{x^2} \).

Step 4: Simplify Each Component

Calculate the square root of each part: \( \sqrt{4} = 2 \) and \( \sqrt{x^2} = |x| \). The absolute value ensures the result is non-negative, matching the characteristic of square roots.

Step 5: Combine the Results

Combine the simplified components to obtain the final result: \( \sqrt{4x^2} = 2|x| \). This is the simplified form of the original expression.

Key Concepts

Properties of Square Roots

Square roots are mathematical functions that "undo" squaring, helping to find out which number, when multiplied by itself, equals the radicand (the number under the square root symbol). There are a few important properties of square roots to bear in mind:

• For any non-negative number, the square root results in a non-negative number.
• The property \( \sqrt{a^2} = a \) is valid if \( a \geq 0 \). This shows how the square root function and squaring are inverses of each other.
• If \( a < 0 \), \( \sqrt{a^2} = |a| \) because square roots inherently yield non-negative results.
• The multiplication property, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \), allows breaking down complex square roots into simpler components.

By utilizing these properties, expressions like \( \sqrt{4x^2} \) can be simplified more easily. For instance, \( \sqrt{4x^2} \) can be split to \( \sqrt{4} \cdot \sqrt{x^2} \).

Absolute Value

The absolute value of a number is essentially its distance from zero on the number line. Absolute value is always non-negative.

• For a positive number \( x \), \(|x| = x\).
• For a negative number \( x \), \(|x| = -x\).

In the context of square roots, absolute value comes into play when dealing with variables. This is because the square root operation always produces a non-negative outcome. For instance, when simplifying \( \sqrt{x^2} \), the expression becomes \(|x|\). This ensures that even if \( x \) was initially negative, the result reflects the non-negative nature of square roots.
Remember, the presence of absolute value keeps expressions consistent with the mathematical properties that dictate the positivity of square roots.

Expression Simplification

Expression simplification is the process of rewriting an expression in a simpler form without changing its value. It often makes mathematical expressions easier to interpret or solve. Here's how this applies to square roots:

• Identify and use mathematical properties such as those of square roots to break down complex expressions.
• Isolate square roots using properties like \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• Simplify the components. For \( \sqrt{4x^2} \), break it down to \( \sqrt{4} \cdot \sqrt{x^2} \), which simplifies further to \( 2 \cdot |x| \).
• Ensure the final expression accurately reflects the properties of the mathematical operations involved, such as using absolute value appropriately.

This process ultimately transforms a complex or messy expression into a more straightforward and understandable format. By practicing expression simplification, you deepen your understanding of mathematical operations and their interrelations.