Problem 36 - Prealgebra

Question

Use the associative property to rewrite each of the following expressions, and then simplify as much as possible. $$\frac{1}{4}(4 x)$$

Step-by-Step Solution

Answer

The simplified expression is \( x \).

Step 1: Identify the Use of the Associative Property

The associative property states that the grouping of numbers can be changed without changing the result. In this expression, we have \( \frac{1}{4}(4x) \), and we can rewrite it as \( (\frac{1}{4} \times 4)x \) by changing the grouping of the numbers.

Step 2: Simplify the Expression Inside the Parentheses

Now, calculate the expression inside the parentheses: \( \frac{1}{4} \times 4 \). This operation results in \( 1 \) because multiplying a number by its reciprocal yields 1.

Step 3: Simplify the Resulting Expression

We now have \( 1 \times x \). Since multiplying any number by 1 results in that same number, the expression simplifies to \( x \).

Key Concepts

Understanding Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. Unlike equations, which are statements that assert two expressions are equal, expressions do not have an equals sign. For the exercise provided, the expression to work with is \( \frac{1}{4}(4x) \). This means we are dealing with fractions, multiplication, and a variable all in one expression. Mathematical expressions like these can look complex at first glance, but using mathematical properties, they can often be rewritten and simplified to become more manageable. Breaking down expressions through simplification or by applying relevant mathematical properties is key to solving them. This process aids in understanding the operations in play and how components within the expression relate to one another.

The Art of Simplification

Simplification is the process of reducing an expression to its most basic form without changing its value. Simplifying an expression makes it easier to handle and understand. In the exercise, simplification begins with identifying that the multiplication inside the parentheses, \( \frac{1}{4} \times 4 \), can be simplified.

• Multiplying a fraction by its reciprocal, such as \( \frac{1}{4} \times 4 \), essentially results in 1.
• After simplifying inside the parentheses, the expression becomes \( 1 \times x \).
• Finally, \( 1 \times x \) simplifies further to just \( x \).

Using simplification, complex expressions can be reduced to simpler terms that reveal more about the value and behavior of the expression itself.

Navigating the Properties of Operations

Properties of operations are mathematical truths that hold for various arithmetic operations. Understanding and applying these properties can greatly assist in simplification and solving of mathematical expressions.One key property used in the provided exercise is the **associative property**. This property applies to both addition and multiplication and states that the way in which numbers are grouped does not affect their sum or product. The associative property allows us to regroup numbers for convenience, but not to change the order of numbers. For example, in multiplication, the property states: \[(a \times b) \times c = a \times (b \times c)\]In the given problem, this was utilized to regroup \( \frac{1}{4} \) and 4 in the expression \( \frac{1}{4}(4x) \) as \((\frac{1}{4} \times 4)x \), making it clear that the multiplication within the parentheses could be evaluated first. Recognizing and leveraging such properties simplifies the task of working with expressions, making them less daunting and more intuitive to simplify.