Problem 10 - Beginning and Intermediate Algebra

Question

Use an associative property to complete each statement. See Example 2. \(3 \cdot(x y)=\)_______

Step-by-Step Solution

Answer

(3 \cdot x) \cdot y

Step 1: Understanding the Associative Property

The associative property states that the way in which numbers are grouped in multiplication does not change their product. For example, in multiplication, \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\). The numbers can be grouped in any order, and the product will remain the same.

Step 2: Applying the Associative Property

Given the expression \(3 \cdot (x \cdot y)\), we apply the associative property. According to this property, we can rearrange the grouping without changing the result. Hence, \(3 \cdot (x \cdot y)\) is equivalent to \((3 \cdot x) \cdot y\).

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in mathematics, representing the idea of scaling or repeated addition. It's like adding a number to itself a certain number of times.
For example, multiplying 3 by 4, written as \(3 \times 4\), means adding 3 four times: 3 + 3 + 3 + 3, which equals 12.

• Multiplication is often shown by symbols such as \(\times\), \(\cdot\), or parentheses, like \(a \cdot b\) or \(ab\).
• This operation is commutative, meaning \(a \cdot b = b \cdot a\). The order of numbers doesn’t affect the result.
• The identity element of multiplication is 1. Any number multiplied by 1 remains unchanged: \(a \cdot 1 = a\).

In exercises involving the associative property, multiplication's flexibility in grouping helps simplify or rearrange expressions without altering the outcome.

Grouping

Grouping within mathematics, especially in multiplication, refers to the organization of numbers to simplify calculations or apply certain properties. An essential property related to grouping is the associative property.
The associative property indicates that the grouping of numbers in multiplication can be rearranged freely. This is expressed as \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\).

• No matter how the numbers are grouped, the result will be the same.
• The associative property supports flexibility in solving mathematical problems, allowing us to rearrange and solve equations more conveniently.

In practical terms, this makes calculations easier, as you can choose the order of operations that simplifies the steps needed to find a solution.

Mathematics

Mathematics is a universal language that describes and analyzes relationships, changes, and quantities. Central to mathematics are the operations and properties that serve as building blocks for more complex concepts.
Within the wide scope of mathematics, fundamental operations like addition, subtraction, multiplication, and division are essential tools used to solve problems and understand patterns.

• Properties like the associative, commutative, and distributive laws enable mathematicians to manipulate and simplify expressions systematically.
• These properties are not just theoretical tools but practical ones, aiding in problem-solving and logical reasoning.
• Understanding these properties helps create a strong mathematical foundation, enabling one to tackle more advanced topics with confidence.

By grasping these principles, students can improve their mathematical skills and enhance their problem-solving capabilities across different areas of study.