Problem 34 - Elementary and Intermediate Algebra

Question

Use the associative law of multiplication to write an equivalent expression. $$ (13 x) y $$

Step-by-Step Solution

Answer

13(xy)

Step 1 - Understand the Associative Law of Multiplication

The associative law of multiplication states that changing the grouping of the factors does not change the product. Mathematically, it is expressed as \((a \times b) \times c = a \times (b \times c)\).

Step 2 - Identify the Factors in the Given Expression

In the expression \((13x) y\), the factors are \(13x\) and \(y\).

Step 3 - Apply the Associative Law of Multiplication

Rearrange the factors using the associative law: \((13x) y = 13 (x y)\).

Key Concepts

mathematical properties

Mathematics includes a variety of properties that simplify operations. One important property is the associative law, particularly in multiplication. The associative law of multiplication states that no matter how factors are grouped, the product remains unchanged. For example, taking numbers and variables: \(a, b, \text{and} c\), the associative law states that: \((a \times b) \times c = a \times (b \times c)\). This law is not just useful for numbers but also applies to algebraic expressions.

algebraic expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. It doesn't have an equals sign. In our exercise, \(13x) y\) is an algebraic expression. Each piece, \(13x\ and \ y\), is called a factor. By using algebraic expressions, we generalize numbers to represent broader concepts. This helps us perform operations like multiplication in a more fluid manner. In the given problem, we reorganize the multiplication without changing the result.

factor grouping

Factor grouping involves the way we organize elements in a multiplication operation to simplify or solve an equation. In the exercise, we start with the expression \(13x) y\). Using the associative law, we regroup the factors as: \(13 (x \times y)\). Grouping factors correctly makes complex problems easier. Understanding factor grouping is vital for solving higher-level algebraic problems efficiently.