Problem 43 - Prealgebra

Question

Simplify each expression. $$7 \cdot(d \cdot 4)$$

Step-by-Step Solution

Answer

The simplified expression is \(28d\).

Step 1: Define the Original Expression

The given expression to simplify is \(7 \cdot (d \cdot 4)\). Our goal is to simplify this expression by applying the associative property of multiplication.

Step 2: Apply the Associative Property of Multiplication

Recall that the associative property of multiplication allows us to group numbers differently without changing their product. We can rewrite the expression \(7 \cdot (d \cdot 4)\) as \((7 \cdot 4) \cdot d\).

Step 3: Multiply Constants Together

First, multiply the constants 7 and 4 together. Calculate \(7 \times 4 = 28\). Thus, the expression becomes \(28 \cdot d\).

Step 4: Write the Simplified Expression

Since there are no further operations to perform, the simplified expression is \(28d\).

Key Concepts

Simplification of Expressions

Simplification of expressions involves making a mathematical expression easier to work with. This often means reducing it to its simplest form. The "simplest form" generally means that there are no more terms that can be combined or operations that can be performed to further compact it.
In the expression provided, namely \(7 \cdot (d \cdot 4)\), the goal was to make it as straightforward as possible.
By applying the associative property of multiplication, we altered the grouping of numbers to simplify the multiplication without changing the result.

• First, we rearranged the numbers and variables: \(7 \cdot (d \cdot 4)\) to \((7 \cdot 4) \cdot d\).
• Next, we performed multiplication of the numbers: \(7 \times 4 = 28\).
• Lastly, we wrote the product with the variable: \(28d\).

Simplifying expressions helps to make computations easier, especially when dealing with more complex mathematical problems later on.

Prealgebra Concepts

Prealgebra lays the foundation for understanding key mathematical ideas and operations. It encompasses the basic principles crucial for solving problems and understanding more complex algebraic concepts.
Prealgebra often involves understanding arithmetic operations like addition, subtraction, multiplication, and division using numbers and variables. These operations can then be expanded through properties like the associative, commutative, and distributive properties to handle expressions more efficiently.
In prealgebra, we use variables, like \(d\) in our example, to represent numbers. This prepares students for more advanced algebra where solving for variables becomes crucial.
The simplification in this exercise demonstrates how prealgebra skills can be used to streamline processes by combining constants and rearranging terms. Keeping these basic principles in mind helps students to lay a strong groundwork for later math courses.

Mathematical Properties

Mathematical properties are rules that help simplify expressions and solve equations efficiently. The associative property is one such property that deals with the grouping of numbers. It states that how numbers are grouped does not affect the product or sum.
For example:

• For multiplication, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
• For addition, \((a + b) + c = a + (b + c)\)

In the exercise, we utilized the associative property to change the grouping from \(7 \cdot (d \cdot 4)\) to \((7 \cdot 4) \cdot d\), simplifying calculations by focusing first on multiplying constants, without affecting the outcome.
Understanding and applying mathematical properties like associative, commutative, and distributive properties, can greatly enhance problem-solving skills and streamline calculation processes.