Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$(2 x+7) 4$$

Step-by-Step Solution

Answer

The expression \( (2x+7)4 \) rewrites to \( 8x + 28 \) when the distributive property is applied

Step 1: Identify the expression

The given expression is \( (2x+7)4 \). Now, using the distributive property i.e, a(b + c) = ab + ac

Step 2: Application of the Distributive Property

By applying the distributive property, we multiply 4 (which is outside the parentheses) with each of the elements inside the parentheses. Thus, it becomes \( 4*2x + 4*7 \)

Step 3: Simplify the equation

After performing the multiplication, the expression simplifies to \( 8x + 28 \)

Key Concepts

Algebraic Expressions

An algebraic expression is like a sentence made up of numbers, variables, and operators. Think of it as a mathematical statement that involves variables such as \(x\) or \(y\), numbers, and operation signs like addition, subtraction, multiplication, and division.
In our example, we start with the algebraic expression \((2x + 7)4\), where \(2x + 7\) is an expression within the parentheses and \(4\) is the factor outside. Here, \(x\) represents a variable, which can vary or change, while \(2\) and \(7\) are constants that stay the same.
The purpose of rewriting algebraic expressions often involves simplifying them to make calculations easier or to see relationships between different mathematical entities better.

Simplifying Expressions

Simplifying expressions is a key part of algebra that makes mathematical equations easier to work with. When simplifying, the goal is to combine like terms or eliminate unnecessary parts of an expression without changing its overall value.
In the exercise given, the operation of simplifying started with the expression \((2x + 7)4\). The expression was simplified using the distributive property to turn it into \(8x + 28\). This simplified expression has no parentheses and each term is expressed clearly.
In practice, always ensure you:

• Multiply terms outside of the parentheses with those inside.
• Look for and combine like terms if possible.
• Reduce the expression to its simplest form.

Simplification often involves multiple steps, and becoming comfortable with each can make you more proficient in working with algebraic expressions.

Mathematical Operations

Mathematical operations are the building blocks of solving equations and expressions. The basic operations include addition, subtraction, multiplication, and division. Understanding how these operations interact, especially in expressions with variables, is crucial for mastering algebra.
In the given exercise, the main operation used is multiplication, as we see in the distributive property where the number \(4\) is multiplied with each term inside the parentheses: \(4 * 2x\) and \(4 * 7\). Knowing how to handle these operations correctly helps you simplify expressions and solve equations accurately.
When performing mathematical operations on expressions:

• Always follow the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Apply operations systematically to maintain the expression’s integrity.
• Double-check your steps to avoid errors, ensuring each operation's application is clear and correct.

Becoming proficient in mathematical operations is crucial because it underpins your ability to manipulate and solve complex algebraic expressions.