Problem 13 - Prealgebra

Question

Simplify each expression. \((3 \cdot w) \cdot 9\)

Step-by-Step Solution

Answer

The simplified expression is \(27w\).

Step 1: Identify the Expression

We are given the expression \((3 \cdot w) \cdot 9\). The goal is to simplify this expression by using properties of arithmetic.

Step 2: Use Associative Property of Multiplication

The associative property states that the way in which numbers are grouped in multiplication does not change the product. Thus, we can regroup the expression as \(3 \cdot (w \cdot 9)\).

Step 3: Simplify the Expression

Now that we have grouped \(w\) and \(9\) together, rewrite the expression as \(3 \cdot w \cdot 9\). Compute \(3 \cdot 9 = 27\).

Step 4: Final Simplified Expression

The expression can now be simplified by performing the multiplication. This gives \(27 \cdot w\), or simply written as \(27w\).

Key Concepts

Understanding Arithmetic Expressions

Arithmetic expressions are a fundamental part of mathematics. They allow us to use numbers and variables in various operations. An arithmetic expression consists of:

• Numbers (also called constants). You see this in our example with the numbers 3 and 9.
• Variables. These are often represented by letters, like 'w' in this case. Variables can change in value.
• Operations like addition, subtraction, multiplication, and division.

In the given expression \((3 \cdot w) \cdot 9\), you see multiplication as the main operation. Understanding these components can help you identify what each part of an expression is doing and how you might simplify or manipulate it. Keep in mind that expressions can be more complex, involving multiple operations and parentheses, which indicate which operations should be performed first.

Simplification of Expressions

Simplification is about making an expression easier to understand without changing its value. In math, this often means combining like terms or using properties to rewrite an expression in a simpler form.To simplify \((3 \cdot w) \cdot 9\), follow these basic steps:

• Regroup numbers and variables using properties like the associative property.
• Perform calculations when possible, like multiplying constants.
• Write the final simplified form, that often looks cleaner and more intuitive.

In our example, simplifying leads us from the original grouping to \(27 \cdot w\). Simplification fosters clarity and makes further calculations easier.

The Role of Mathematical Properties

Mathematical properties are rules that always hold true in arithmetic operations. They guide us in how to manipulate expressions effectively and are particularly handy in simplifying expressions.One crucial property used in this example is the Associative Property of Multiplication. It states that no matter how you group together the numbers (or variables), the product remains the same. For example, in\((3 \cdot w) \cdot 9\) and \(3 \cdot (w \cdot 9)\), you'll still get the same result.Using properties like these, we can:

• Reorder or regroup terms to make calculations friendlier.
• Understand deeper relationships between numbers and variables.
• Simplify expressions efficiently, as demonstrated in changing \((3 \cdot w) \cdot 9\) to \(27w\).

Learning and applying these properties improve your mathematical fluency and problem-solving skills.