Problem 45 - Algebra 2

Question

Write an algebraic expression to represent each verbal expression. the cube of the difference of a number and 7

Step-by-Step Solution

Answer

The algebraic expression is \((x - 7)^3\).

Step 1: Identify the Verbal Expression

The verbal expression given is "the cube of the difference of a number and 7." This means we must first determine what the number is and then find the cube of that result.

Step 2: Translate 'A Number' to a Variable

In algebra, we often use a letter to represent an unknown number. Let's use the variable \( x \) to represent this unknown number.

Step 3: Express 'The Difference of a Number and 7'

The phrase "the difference of a number and 7" can be expressed in algebraic terms as \( x - 7 \). This means we subtract 7 from the variable \( x \).

Step 4: Find 'The Cube of the Difference'

To find the cube of \( x - 7 \), we need to raise the expression to the power of 3. So the cube of \( x - 7 \) is \((x - 7)^3\).

Step 5: Write the Final Algebraic Expression

The final algebraic expression representing the verbal expression "the cube of the difference of a number and 7" is \((x - 7)^3\).

Key Concepts

Verbal Expressions

Verbal expressions are phrases or sentences that describe mathematical operations using words instead of symbols. They are like instructions for turning words into math. To create a bridge between language and mathematics, we first need to understand what is being said. For example, in the phrase "the cube of the difference of a number and 7," we break it down into parts:

• "The difference of a number and 7" tells us there is a subtraction operation involved.
• "The cube of" indicates we are raising something to the power of 3.

This translation is key in algebra, helping us convert words into expressions that we can manipulate mathematically. By doing this, we begin to see how language can describe complex mathematical concepts. Understanding verbal expressions is crucial for solving word problems and creating algebraic equations from them.

Variable Representation

In algebra, a variable is a symbol or letter that stands in for an unknown value. It acts as a placeholder for a number we don't know yet. This allows us to write general expressions that can apply to many situations. In our example, we chose the letter "\( x \)" as the variable.

• The phrase "a number" usually signifies that a variable is needed.
• We often use letters like \( x \), \( y \), or \( z \) to represent these unknowns.

By using variables, we can create expressions that are flexible and adaptable. This is because variables can be replaced with different numbers depending on the situation. Having a good grasp of variable representation helps in setting up and solving algebraic equations easily.

Operations in Algebra

Operations in algebra use symbols to perform basic mathematical processes such as addition, subtraction, multiplication, and division. In any verbal expression, identifying and correctly performing these operations is crucial. For instance, the operation to find "the difference of a number and 7" involves subtraction (\( x - 7 \)). Here, you subtract 7 from the variable \( x \), which shows how operations work in tandem with variables.

• Subtraction: Indicated by terms like "minus," "less," or "difference."
• Addition: Described by words such as "plus" or "more than."
• Multiplication: Key terms include "times" or "product."
• Division: Phrases might be "divided by" or "quotient."

Understanding these operations is foundational to mastering algebra, as they form the building blocks for constructing expressions and equations.

Exponents and Powers

Exponents and powers allow us to express repeated multiplication compactly. They are a shortcut for multiplying a number by itself several times. In our problem, the phrase "the cube of" translates to raising the expression to the power of 3. When you see "cube," think of this operation:

• Cubing (raising to the third power): Refers to multiplying a number by itself two more times, e.g., \( (x - 7)^3 = (x - 7) \times (x - 7) \times (x - 7) \).

Understanding exponents helps in simplifying math tasks that involve large multiplications. It's also pivotal for working on polynomial equations and understanding the behaviors of functions involving powers. Mastery of exponents and powers allows one to recognize and manipulate mathematical expressions efficiently.