Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of addition to write an equivalent algebraic expression. $$6(x+4)$$

Step-by-Step Solution

Answer

The equivalent algebraic expression using the commutative property is \(6x+24\).

Step 1: Identify the terms inside the parenthesis

The terms inside the parentheses here are 'x' and '4'.

Step 2: Apply distributive property

Then, distribute '6' to every term inside the parentheses, which gives \(6*x + 6*4\).

Step 3: Simplify the expression

Simplify the expression to obtain \(6x+24\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations to represent a mathematical relationship or formula. These expressions can include addition, subtraction, multiplication, division, and exponents, following the conventional rules of arithmetic. In the context of the exercise, we have the expression \(6(x+4)\), which includes a number \(6\), a variable \(x\), and a constant \(4\), combined using multiplication and addition. Understanding how to work with such expressions is fundamental in algebra and helpful in solving various mathematical problems.

When faced with algebraic expressions, it's crucial to identify the terms and operations involved before applying properties like the commutative property of addition, which allows us to rearrange the terms freely when adding. This property ensures consistency in our calculations and simplification processes.

Distributive Property

The distributive property is a cornerstone of algebra that enables us to simplify expressions by distributing a factor across terms inside a parenthesis. Expressed as \(a(b+c)=ab+ac\), it allows us to multiply a single term by each term in a binomial or polynomial. In our exercise, applying the distributive property to the expression \(6(x+4)\) involves multiplying \(6\) by \(x\) and \(4\) separately, resulting in \(6*x + 6*4\).

This property is especially useful as it applies to all real numbers, variables, and algebraic expressions, making it a powerful tool for simplifying complex problems. It's also essential when we can't perform addition or subtraction first due to lack of like terms or when multiplication across an addition or subtraction is required.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in their most reduced form, making them easier to understand or solve. This involves combining like terms, using properties of operations, and sometimes factoring expressions. In this exercise, after applying the distributive property, we should simplify the expression \(6*x + 6*4\) by performing the multiplication, which yields \(6x + 24\).

Simplification is a critical step in problem-solving as it can reveal the structure of an algebraic expression or equation, help in identifying possible solutions, or make further manipulations more manageable. It's also a way to make sure that our expression is as tidy and concise as possible before proceeding with any other algebraic operations.

Algebraic Operations

Algebraic operations refer to the various procedures we can apply to algebraic expressions, such as addition, subtraction, multiplication, division, and the application of exponentiation rules. These operations allow us to manipulate and solve algebraic expressions and equations. In the given exercise, we primarily focus on multiplication and addition. We use multiplication to apply the distributive property and addition to combine terms.

Mastering these operations is critical for solving algebraic problems efficiently and accurately. They form the basis for more advanced topics in algebra and calculus, where understanding and applying them correctly becomes even more critical. Being able to move fluidly from one operation to another, ensuring accuracy at every step, is an essential skill in mathematical problem-solving.