Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Express each sentence as a single algebraic expression. Then simplify the expression. Cube a number. Subtract 6 from this exponential expression. Multiply this difference by 4

Step-by-Step Solution

Answer

The simplified expression of the phrase 'Cube a number. Subtract 6 from this cube. Multiply this difference by 4' where number is \(x\) is \(4x^3 - 24\).

Step 1: Translate the Words into Algebraic Expression

Starting with the number, which we'll call \(x\), the exercise says to cube it, resulting in \(x^3\). Then, we subtract 6 from this cube, giving us \(x^3 - 6\). Finally, it instructs us to multiply this difference by 4, meaning our entire expression is \(4(x^3 - 6)\).

Step 2: Simplify the expression

The expression \(4(x^3 - 6)\) could be expanded and simplified. The multiplication distributes over the subtraction, resulting in \(4x^3 - 24\) which is the simplest form of our expression.

Key Concepts

Cubing a Number

Understanding the process of cubing a number is fundamental in algebra. Cubing refers to raising a number to the power of three, which is written in exponential form as the number followed by a superscript 3, for example, if we have a number represented by the variable x, when we cube it, we write it as x^3. This expression represents x multiplied by itself three times: x * x * x.

Students often encounter problems that involve cubed numbers when working with volumes of cubes, since volume is calculated by multiplying the length, width, and height of a cube—essentially cubing the side length. Moreover, cubing functions are a type of polynomial function with interesting curves, often used to model real-world phenomena in physics, engineering, and economics.

The operation of cubing is also pertinent to the understanding of algebraic manipulation and simplification, as it sets the stage for more advanced operations and problem-solving strategies.

Simplifying Expressions

Simplifying algebraic expressions makes them easier to work with by reducing them to their simplest form. When an expression involves several mathematical operations, such as exponentiation, multiplication, and subtraction, following the correct order of operations is crucial. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is a common mnemonic to help remember this sequence.

Let's consider the expression from our exercise: 4(x^3 - 6). Simplification starts by applying the distributive property, which entails multiplying the 4 by each term inside the parentheses. The expression thus becomes 4x^3 - 24, where 4x^3 stands for '4 times the cubed number' and -24 is the product of the number 4 and -6.

It's crucial to follow these steps meticulously, as simplification not only makes expressions easier to grasp but also prepares them for solving equations and inequalities and for graphing functions.

Translating Word Problems to Algebra

One of the most valuable skills in algebra is translating word problems into algebraic expressions. Students often face the challenge of converting a problem's verbal description into a form that can be analyzed mathematically. To do so effectively, identifying and defining variables is the first step. Variables are symbols, typically letters, that represent unknown values.

Once variables are defined, the next phase is constructing an expression based on the actions described in the problem. For instance, 'cube a number and subtract 6' leads to an initial expression of x^3 - 6 if x represents the number in question. Finally, operations such as 'multiply this difference by 4', further refine the expression into 4(x^3 - 6).

With practice, students can develop an intuitive sense of how to bridge the gap between words and algebraic language, a skill that is useful not only in mathematics but in fields where quantitative analysis of text-based information is required. Presenting algebra in a relatable context, through word problems, helps demystify the subject and showcases its widespread applicability.