Problem 95 - Introductory Algebra for College Students

Question

In Exercises \(93-96\), let \(x\) represent the number. Express each sentence as a single algebraic expression. Then simplify the expression. Cube a number. Subtract 4 from this exponential expression. Multiply this difference by \(5 .\)

Step-by-Step Solution

Answer

The simplified algebraic expression that corresponds to the sentences given in the exercise is \(5x^3 - 20\).

Step 1: Cube the number

The first sentence says to cube a number. In terms of algebraic expression, this translates to \(x^3\), where \(x\) is the number given.

Step 2: Subtract 4 from the cube

The next sentence says to subtract 4 from the number cubed. So, that expression can be written as \(x^3 - 4\).

Step 3: Multiply the expression by 5

The final sentence requires the result from the previous step to be multiplied by 5. The expression to this should look like \(5(x^3 - 4)\).

Step 4: Simplify the expression

Next, distribute the 5 across the terms in the parentheses, which results in the final simplified expression \(5x^3 - 20\).

Key Concepts

Basic Algebra Concepts

Basic algebra is like the first step into the world of mathematical problem-solving. It's the foundation that enables us to translate real-world situations into mathematical language. At the core, algebra involves using letters and symbols such as \(x\) to represent numbers or quantities that can change. In our original exercise, \(x\) is used to represent an unknown number which we'll cube, manipulate, and simplify.

In algebra, operations like addition, subtraction, multiplication, and division are performed on these variables, just like numbers. Thus, understanding how to handle these operations with both numbers and symbols is crucial. Basic algebra isn't just about solving for \(x\). It's about knowing what \(x\) means, how it interacts with numbers, and how we can express relationships between quantities efficiently.

Mastering basic algebra means being comfortable with expressions. Eventually, this skill makes even complex problems easier to untangle.

Simplification Techniques

Simplification in algebra is all about making expressions easier to handle. After setting up our initial algebraic expression, we usually simplify it to make the solution clearer and more manageable. In our example, the initial expression \(5(x^3 - 4)\) can look daunting, but the process of simplification involves familiar steps.

• **Distribute:** Take the term outside the parentheses and multiply it by each term inside. Here, multiplying \(5\) by both \(x^3\) and \(-4\) results in the simplified expression \(5x^3 - 20\).
• **Combine Like Terms:** Although not necessary in this exercise, combining like terms is crucial when there’s more than one variable or term.
• **Factor:** Sometimes, expressions are simplified by factoring out common elements, but that's not required with \(5x^3 - 20\).

Simplification turns a complex expression into something more readable. It's like clearing away the clutter to see the core relationships between numbers and variables.

Understanding Mathematical Expressions

Mathematical expressions are quite literally the language of mathematics. They condense information into a manageable form. In exercises involving expressions, you often start with a sentence or a situation, much like our problem, and then transform it into an expression like \(5(x^3 - 4)\).

Expressions do not equate to anything on their own; that's the job of an equation. They’re often like sentences missing a verb—indicating a relationship without giving a solution. In our example, the transformation of words into \(5(x^3 - 4)\) involves understanding keywords: 'cube' translates to \(x^3\), and 'subtract' shows up as \(-4\).

• **Terms:** Expressions consist of terms (e.g., \(5x^3\) and \(-20\)), which can be constants or consist of variables.
• **Operators:** Signs like \(+\) or \(-\) that indicate operations between terms.

Understanding what makes up an expression helps in manipulating and simplifying it to suit the problem’s needs, leading to effective problem-solving in mathematics.