Problem 78 - 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 given expression is demonstrating the Commutative Property of Multiplication.

Step 1: Simplify the First Part of the Equation

Using the Multiplication Operation on the left side of the equation, \(6 \cdot(2 \cdot 3)\) simplifies to \(6 \cdot 6\) which equals \(36\).

Step 2: Simplify the Second Part of the Equation

Using the Multiplication Operation on the right side of the equation, \(6 \cdot(3 \cdot 2)\) simplifies to \(6 \cdot 6\), which equals \(36\).

Step 3: Identify the Mathematical Property

Comparing both results, it is seen that the order of multiplication does not change the result, implying that the property illustrated is the Commutative property of multiplication.

Key Concepts

Properties of Multiplication

Multiplication is one of the fundamental operations in mathematics, and it comes with a set of properties that make it easier to work with numbers. These properties are essential because they help simplify equations and solve mathematical problems more efficiently. One of these is the **Commutative Property**.The commutative property tells us that the order in which you multiply numbers does not affect the product. In simple terms, if you switch the numbers, the result remains the same. For example:

• The expression \(2 \cdot 3 = 6\) is the same as \(3 \cdot 2 = 6\). The product is the same no matter the order of numbers.
• Similarly, \(6 \cdot(2 \cdot 3)\) is the same as \(6 \cdot(3 \cdot 2)\), both of which equal \(36\).

Understanding and using the commutative property is especially useful when simplifying complex algebraic expressions or when performing quick mental calculations.

Equation Simplification

Equation simplification is a key skill in algebra and mathematics as a whole. It lets you transform expressions into simpler forms, making it easier to solve or compare them. When we simplify an equation, we aim to reduce it to the simplest possible terms without changing its value.In the example exercise, you see equation simplification in action. We begin by simplifying each part of the expression separately:

• Simplify \(6 \cdot(2 \cdot 3)\) to get \(36\).
• In parallel, simplify \(6 \cdot(3 \cdot 2)\) to also get \(36\).

After simplifying both sides, we find that they equate to the same value. Simplifying expressions is not just about making expressions shorter; it's about revealing the underlying properties, like the commutative property, that help solve and understand equations more deeply.

Algebraic Expressions

Algebraic expressions are combinations of numbers, symbols, and operations that represent quantities in a general form. Learning to interpret and manipulate these expressions is a core skill in algebra.Every algebraic expression can be broken down into its parts, including numbers, variables, and operation symbols like plus, minus, multiply, or divide.

• Consider the expression \(6 \cdot(2 \cdot 3)-6 \cdot(3 \cdot 2)\).
• Here, the expression is made up of two multiplications and a subtraction operation.

To solve algebraic expressions, you apply properties of operations, such as the commutative and associative properties. These properties allow us to rewrite expressions in ways that might be more manageable or revealing. By mastering algebraic expressions, you improve your ability to model real-world situations and solve equations that describe them. This knowledge is not only crucial for academic purposes but also for everyday problem-solving.