Problem 5 - Elementary and Intermediate Algebra

Question

Concept Reinforcement Classify each of the following statements as either true or false. The expression \(\sqrt[3]{X}\) is not simplified if \(X\) contains a factor that is a perfect cube.

Step-by-Step Solution

Answer

True. If \X\ contains a factor that is a perfect cube, \(\root[3]{X}\) can be simplified, making the statement true.

Step 1: Understand the Problem

The task is to determine if the statement regarding the simplification of the expression \(\root[3]{X}\) is true or false. Specifically, check if the statement 'The expression \(\root[3]{X}\) is not simplified if \X\ contains a factor that is a perfect cube' is correct.

Step 2: Identify Key Concept

Recognize that an expression involving a cube root, \(\root[3]{X}\), can be simplified further if \X\ contains a factor that is a perfect cube. Simplifying means eliminating or reducing these perfect cube factors.

Step 3: Analyze the Expression

If \X\ contains a perfect cube factor \k^3\, the expression \(\root[3]{X}\) can be simplified to \(\root[3]{k^3} \cdot \root[3]{Y} = k \cdot \root[3]{Y}\), where \(X = k^3 \cdot Y\). This form is simpler than the original.

Step 4: Evaluate the Statement

Given that \(X\) containing a perfect cube factor implies \(\root[3]{X}\) can be simplified, the statement 'The expression \(\root[3]{X}\) is not simplified if \X\ contains a factor that is a perfect cube' is therefore true.

Key Concepts

Perfect Cube Factor

In algebra, a perfect cube is a number that can be written as the cube of an integer. For instance, 8 is a perfect cube because it can be expressed as \(2^3\), and 27 is a perfect cube because it can be written as \(3^3\). Identifying perfect cube factors is crucial when simplifying cube roots.

Let's consider the expression \(\sqrt[3]{X}\). If \(X\) contains a perfect cube factor, it is possible to simplify the expression further. For example, if \(X = 27\), then we can write \(\sqrt[3]{27} = \sqrt[3]{3^3} = 3\).

Whenever you spot that \(X\) contains a factor that is a perfect cube, it signals that the expression can be simplified. Simplifying helps in making complex mathematical expressions easier to handle and understand.

Cube Root Simplification

Cube root simplification involves reducing an expression containing a cube root by eliminating perfect cube factors.

Here's how you can simplify cube roots step-by-step:

• First, factorize \(X\) to identify any perfect cube factors.


• Next, separate the perfect cube factor from the rest of the expression.


• Simplify the perfect cube factor outside the cube root.

For example, if you have \(\sqrt[3]{54}\), you can factorize \(54\) as \(2 \times 27\). Since \(27\) is a perfect cube \(\big(3^3\big)\), you can simplify the expression to \(3 \times \sqrt[3]{2}\). The simplification process makes the expression \(\sqrt[3]{54} = 3 \sqrt[3]{2}\) much simpler and more manageable.

Simplifying cube roots not only makes the equations easier to handle but also often provides more insight into the nature of the mathematical relationship.

Expression Analysis

Expression analysis is the process of evaluating whether an algebraic expression, such as one involving a cube root, can be simplified. This evaluation is based on identifying factors and recognizing potential simplifications.

In the context of cube roots, expression analysis begins by determining if \(X\) contains any perfect cube factors. Once identified, these factors allow for the simplification of the cube root expression.

Take for instance the statement: 'The expression \(\sqrt[3]{X}\) is not simplified if \(X\) contains a factor that is a perfect cube.' To analyze this, identify the perfect cube factors within \(X\). As seen earlier, the expression \(\sqrt[3]{54}\) can be simplified to \(3 \times \sqrt[3]{2}\), proving that the statement is true.

Through expression analysis, one confirms that an unsimplified cube root expression always has the potential for simplification if it contains perfect cube factors. This insight is fundamental to mastering algebraic manipulations and simplifying complex problems.