Problem 32 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(x^{3}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Power of a Power Property

To simplify the expression \(left(x^{3}\right)^{1 / 3}\), begin by applying the property of exponents \((a^{m})^{n} = a^{m cdot n}\). In this case, \(a = x\), \(m = 3\), and \(n = \frac{1}{3}\). Multiply the exponents:\[x^{3 cdot \frac{1}{3}}\]

Step 2: Simplify the Exponent

Now, simplify the exponent \(3 cdot \frac{1}{3}\). The multiplication yields:\[3 cdot \frac{1}{3} = 1\]

Step 3: Simplify the Expression

After simplifying the exponent, the expression becomes:\[x^{1}\] This simply means \(x\).

Step 4: Consider the Absolute Value

Since \(x\) can be any real number, consider the absolute value for the simplification:\[|x|\]This ensures the result is non-negative, as required by absolute value constraints when dealing with real numbers.

Key Concepts

Power of a Power Property

The power of a power property is a useful rule when simplifying expressions involving exponents. This property states that when you have an exponent raised to another exponent, like \((a^m)^n\), you can simplify it by multiplying the exponents together: \(a^{m \cdot n}\).

Let’s break this down with an example provided in the exercise. Consider \((x^3)^{1/3}\). Here, \(x\) is the base, \(3\) is the first exponent \(m\), and \(1/3\) is the second exponent \(n\). By applying the power of a power property, you multiply the two exponents to get \(x^{3 \cdot (1/3)} = x^1\).

This simplification shows how powerful this rule can be for reducing complex exponent expressions into simpler forms. It saves time and helps deal with complex algebraic functions by focusing on algebraic properties rather than performing lengthy calculations.

Absolute Value

Absolute value is used to denote the non-negative value of any real number. It ensures that even if a number is negative, its absolute value represents its magnitude in a non-negative form, denoted by two vertical bars: \(|x|\).

In the context of the exercise, once we simplified \((x^3)^{1/3}\) to \(x\), it’s crucial to acknowledge that \(x\) could potentially be any real number. This means \(x\) might be negative.

To address this, we use the absolute value, writing \(|x|\) to represent the final simplified expression. This notation guarantees that regardless of whether \(x\) is positive or negative, the outcome remains non-negative. The absolute value ensures our expression remains valid and consistent across all real numbers.

Real Numbers

Real numbers encompass all numbers on the continuous spectrum that can be represented on a number line, including both rational numbers (like fractions and integers) and irrational numbers (such as \(\pi\) and \(e\)).

In the exercise, it's vital to assume that the variable \(x\) can be any real number. This is because real numbers include all possible values \(x\) can take on, whether they be fractions, whole numbers, or decimals with infinite non-repeating sequences.

Addressing \(x\) as a real number means considering all its possible forms under the operations conducted, especially in the context of simplification and conversion through properties such as the power of a power property and absolute value. Emphasizing the real number set highlights the wide application of mathematical principles and ensures solutions are universally suitable for any computation involving \(x\).