Problem 33 - Elementary and Intermediate Algebra

Question

Use the associative law of multiplication to write an equivalent expression. $$ (7 m) n $$

Step-by-Step Solution

Answer

7 (mn)

Step 1: Understand the Associative Law of Multiplication

The associative law of multiplication states that for any three numbers or variables, say a, b, and c, the equation \( (a \times b) \times c = a \times (b \times c) \) holds true. This means the way numbers are grouped in multiplication does not change the product.

Step 2: Identify the Given Expression

The given expression is \( (7m) n \). Here, 7, m, and n are the numbers or variables involved in the multiplication.

Step 3: Apply the Associative Law

Using the associative law, we can group the numbers/variables differently. So, \( (7m) n \) can be written as \( 7 (m \times n) \).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables (like m and n), and operation symbols. In our example, the given expression is \( (7m) n \). Here, 7 is a number, and m and n are variables.

In algebraic expressions, variables can represent unknown values or values that can change. Understanding algebraic expressions is essential for solving equations and simplifying problems.

Multiplication Properties

Multiplication has several important properties that make it easier to work with algebraic expressions. One of the key properties is the Associative Law of Multiplication. This law states that for any three numbers or variables, the grouping of the numbers does not affect the product. Mathematically, \( (a \times b) \times c = a \times (b \times c) \).

This property helps in simplifying complex expressions and making calculations more manageable. In our exercise, we applied the associative law to rewrite \( (7m) n \) as \( 7 (m \times n) \), demonstrating that the grouping of numbers/variables can be altered without changing the product.

Variable Manipulation

Variable manipulation refers to the process of rearranging and simplifying expressions involving variables. In algebra, we often need to manipulate variables to solve equations or simplify expressions.

In our example, the variables m and n are part of the expression \( (7m) n \). By applying the associative law, we changed the grouping to \( 7 (m \times n) \). This is a simple but powerful form of variable manipulation.

Through variable manipulation, we can better understand the relationships between variables and their respective values in different contexts. It is a fundamental skill in algebra that helps in solving and simplifying various kinds of mathematical problems.