Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The equivalent algebraic expression for \(6(x+4)\) using the commutative property of multiplication is \(6x+24\).

Step 1: Understand the Commutative Property

The commutative property states that for any two real numbers, the product remains the same regardless of the order of multiplicands. i.e., for any real numbers 'a' and 'b', a*b = b*a.

Step 2: Apply the Commutative Property to the Expression

We apply the commutative property to this expression \(6(x+4)\) by reordering the multiplication. The expression \(6(x+4)\) is the same as \(x*6 + 4*6\).

Step 3: Simplification

This simplification results in the equivalent expression \(6x + 24\).

Key Concepts

Algebraic Expression

An algebraic expression is a fundamental concept in mathematics. It's a combination of numbers, variables, and operators (like plus and minus). These expressions do not hold an equality sign and can look like this: \(3x + 2\) or \(5a - 7b + c\). Algebraic expressions are the building blocks of equations, playing a crucial role in algebra.
Variables act as placeholders for numbers, which allows expressions to represent different values. For instance, in \(6(x+4)\), \(x\) is a variable.
Numbers in these expressions are known as coefficients. Here, \(6\) is the coefficient of \(x\), showing how many times \(x\) is multiplied.

• Algebraic expressions can be simplified using properties of operations like the distributive property.
• These expressions help in solving real-world problems by forming equations when set equal to values.

Understanding algebraic expressions is essential for solving equations and puzzles in algebra. Without simplification through properties like the distributive property, expressions can seem much more complex.

Distributive Property

The distributive property is another core concept in algebra. It allows you to multiply a number by a sum or difference inside parentheses. The property is expressed as \(a(b + c) = ab + ac\). This means multiplying \(a\) with both \(b\) and \(c\) individually and then adding the results.

• This property simplifies problems by breaking them into easier parts.
• It is especially useful when dealing with algebraic expressions, like \(6(x+4)\).

In the exercise, applying the distributive property means multiplying \(6\) by both \(x\) and \(4\). This simplifies to the expression \(6x + 24\).
Using the distributive property effectively reduces complex multi-layered calculations into simpler arithmetic, helping you see patterns and make connections.

Multiplication in Algebra

Multiplication in algebra is not just arithmetic; it involves variables and expressions, making it a vital operation in mathematics. When you multiply in algebra, you're often dealing with expressions that include numbers and variables, like in \(6(x+4)\).

• Multiplication allows combining terms and simplifying expressions.
• It's often used alongside the commutative, associative, and distributive properties.

In \(6(x+4)\), multiplication involves distributing \(6\) across each term inside the parentheses: \(6\times x\) and \(6\times 4\), resulting in \(6x + 24\).
Understanding how multiplication works when variables are involved is crucial for solving and simplifying algebraic expressions. It enables you to transform expressions into their equivalent forms, making them easier to work with.