Problem 31 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(x^{2}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Power of a Power Rule

The expression given is \( \left(x^{2}\right)^{1/2} \). According to the Power of a Power Rule in exponents, \( \left(a^m\right)^n = a^{m\cdot n} \). Apply this rule here: \( \left(x^{2}\right)^{1/2} = x^{2\cdot (1/2)} = x^1 = x \).

Step 2: Apply Absolute Value Where Needed

Since we are simplifying potentially across all real numbers, consider the requirement for absolute values when dealing with square roots of squared terms. The expression simplifies to \( x \), but since squaring a negative number makes it positive, the expression \( \sqrt{x^2} \) should be \( |x| \). Therefore, the expression simplifies to \( |x| \).

Key Concepts

Absolute Value

When dealing with the concept of absolute value, it's important to understand that it represents the distance of a number from zero on the number line.
Distance is always a non-negative quantity. Therefore, the absolute value of a number is always positive, or zero if the number itself is zero. In mathematical terms, the absolute value of a number \(x\), written as \(|x|\), is described as follows:

• If \(x \, \geq \, 0\), then \(|x| = x\).
• If \(x \, < \, 0\), then \(|x| = -x\), essentially flipping \(x\) to positive.

So when we encounter \(x\) in any expression, especially in the context of simplifying expressions with real numbers, we must consider this property.
For instance, in the expression \(\sqrt{x^2}\), even though it simplifies analytically to \(x\), the correct algebraic simplification is \(|x|\). This ensures the outcome respects the nature of numbers, remaining non-negative regardless of \(x\)'s initial sign.

Power of a Power Rule

This rule is a crucial part of understanding exponentiation in algebra. It provides a method to simplify expressions where an exponent is raised to another power.
The rule is: \(\left(a^m\right)^n = a^{m \cdot n}\).
The usefulness of this rule can be seen when simplifying expressions like \(\left(x^2\right)^{1/2}\). You multiply the exponents together:

• First, calculate the inner exponent: \(x^2\).
• Apply the power of a power rule: \((x^2)^{1/2} = x^{2 \cdot (1/2)}\).
• This results in: \(x^1 = x\).

However, remember that if it involves real numbers, and especially if variables represent real numbers, the simplification should include absolute values.
This is because \(x^1\) alone does not account for possible negative values of \(x\). We need \(|x|\) to ensure the output is always positive.

Real Numbers

Real numbers are a fundamental part of mathematics, representing all the numbers that can be found on the number line.
They include rational and irrational numbers, covering everything from integers, whole numbers, and fractions to decimals that don’t repeat or end. In short, real numbers include:

• Positive integers (like 1, 2, 3, ...)
• Negative integers (such as -1, -2, -3, ...)
• Fractions (examples are 1/2, 2/3, ...)
• Decimals both terminating (like 0.5, 0.75) and repeating (such as 0.333...)
• Irrational numbers (like \(\pi\) and \(\sqrt{2}\))

Real numbers are crucial because they offer a complete scenario of number types that can potentially be utilized in equations and expressions.
When simplifying expressions, such as \(\sqrt{x^2}\), acknowledging that \(x\) can be any real number ensures the use of absolute values, making equations valid across all possible real-valued outcomes.