Problem 69 - Prealgebra

Question

Rewrite each expression using the associative property of addition or multiplication. $$3(4 y)$$

Step-by-Step Solution

Answer

The expression rewritten using the associative property is \((3\cdot4)y = 12y\).

Step 1: Identify the Property

Recognize that the expression involves multiplication and the associative property of multiplication states that the way in which factors are grouped does not change the product. Specifically, for any numbers or variables \(a\), \(b\), and \(c\), \(a(bc) = (ab)c\).

Step 2: Rewrite the Expression

Apply the associative property of multiplication to the expression by changing the grouping of the numbers and variables. The original expression is \(3(4y)\). According to the associative property, it can be rewritten as \((3\cdot4)y\).

Step 3: Simplify the Expression

Multiply the numbers to simplify the expression. Calculate \(3\cdot4\) to get 12, giving you the expression \(12y\). Therefore, the expression \(3(4y)\) can be rewritten as \((3\cdot4)y = 12y\).

Key Concepts

associative property of multiplication

The associative property of multiplication is a fundamental rule in mathematics. It helps us understand how different ways of grouping numbers or variables do not affect the outcome of the multiplication. This property states that when you multiply three or more numbers, the way you group the numbers does not change the product. In mathematical terms, it means that

• \(a(bc) = (ab)c\)

For example, in the expression \(3(4y)\), we are multiplying three components: 3, 4, and \(y\). According to the associative property, we can group the components differently without changing the result. Thus, \(3(4y)\) can also be written as \((3\cdot4)y\).
This flexibility is useful when simplifying expressions or evaluating complicated mathematical problems. By rearranging, we can make calculations easier.

properties of operations

The properties of operations are rules that describe how numbers interact in arithmetic. While there are several properties—such as commutative, associative, and distributive—the associative property is specifically useful for understanding multiplication and addition involving multiple terms.

For multiplication, the associative property allows you to change the grouping of numbers in a product. This is particularly handy when dealing with long, complex expressions as it can simplify calculations. Other operational properties include:

• Commutative property: The order of numbers can be swapped without affecting the result (e.g., \(a \cdot b = b \cdot a\)).
• Distributive property: Describes how multiplication is distributed over addition, such as \(a(b+c) = ab + ac\).

Understanding these properties helps us perform more efficient and error-free calculations, by choosing the most strategic way to group or order numbers and operations.

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value or set of values. Think of them like sentences that tell an arithmetic story. They can appear complicated, but understanding the underlying operations and properties can simplify them.

In the expression \(3(4y)\), the goal is to rewrite it using the associative property of multiplication to make it simpler. By using this property, the expression becomes \((3\cdot4)y\), which simplifies further to \(12y\). This simplification tells us that the product of 3 and 4, multiplied by \(y\), is 12 times \(y\).
Practicing how to rewrite and simplify expressions using different properties allows us to solve problems in mathematics more effectively. Developing this skill is crucial for anyone looking to deepen their understanding of math.