Problem 52 - College Algebra

Question

state the name of the property illustrated. $$ 6 \cdot(2 \cdot 3)=6 \cdot(3 \cdot 2) $$

Step-by-Step Solution

Answer

The property illustrated is the Commutative Property of Multiplication.

Step 1: Analyze the given equation

Given is \(6 \cdot(2 \cdot 3)=6 \cdot(3 \cdot 2)\). Observe carefully that both the sides of the equation are equal, irrespective of the changing order of multiplication within the parenthesis, which means the order of the numbers doesn't affect the multiplication operation.

Step 2: Identify the property

Considering the Commutative Property, the order of the numbers may be switched around and does not affect the outcome of the problem. Therefore, the property of real numbers illustrated in this equation is the Commutative Property of Multiplication.

Key Concepts

Properties of Real Numbers

Understanding the properties of real numbers is essential in simplifying algebraic expressions and solving equations. Real numbers are all the numbers that exist on the number line, including rational and irrational numbers. Their properties are rules that apply to the addition and multiplication of these numbers, and they form the foundation of algebra.

There are several properties of real numbers, but when it comes to multiplication, the most relevant ones are the Commutative Property, Associative Property, and Distributive Property. The Commutative Property indicates that the order in which two numbers are multiplied doesn't affect the result; for example, \( a \times b = b \times a \). Meanwhile, the Associative Property refers to the way numbers are grouped in an expression and asserts that the product remains the same regardless of how the numbers are grouped (e.g., \( (a \times b) \times c = a \times (b \times c) \)). And the Distributive Property relates multiplication and addition, exemplified by \( a \times (b + c) = (a \times b) + (a \times c) \).

These properties become immensely powerful tools in algebra to simplify expressions and to perform operations with ease.

Associative Property

The Associative Property is a rule that pertains to the grouping of numbers. It asserts that the way in which numbers are grouped when adding or multiplying them does not affect the sum or product. For multiplication, the Associative Property can be written as \( (a \times b) \times c = a \times (b \times c) \).

When solving problems that involve multiple numbers, being able to regroup them helps in finding easier ways to calculate. It allows us to 'associate' numbers in a manner that can simplify mental math or paper calculations. For example, if you have \( 2 \times (3 \times 4) \), you can rearrange it to \( (2 \times 3) \times 4 \), making it easier to compute either by calculating 3 times 4 first or 2 times 3 first, whichever you find simpler. The Associative Property is especially useful when dealing with complex algebraic expressions, as it provides flexibility in how terms are combined and multiplied.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (letters that stand for numbers), and arithmetic operations like addition, subtraction, multiplication, and division. They are fundamental components of algebra and can be simple or complex, with multiple terms and variables.

An example of a simple algebraic expression might be \( 3x + 4 \), which expresses a relationship between numbers and a variable. Understanding how to work with these expressions involves applying the properties of real numbers, such as the Commutative and Associative properties, to simplify and solve them.

For instance, if you have the expression \( (x \times 2) \times y \), you can apply the Associative Property to group the terms as \( x \times (2 \times y) \), which might make it easier to compute if the values of \( x \) and \( y \) are known. Algebraic expressions are the foundational building blocks that lead to equations, and learning to manipulate them correctly is essential for success in mathematics and many practical applications.