Problem 35 - Prealgebra

Question

Use the associative property to rewrite each of the following expressions, and then simplify as much as possible. $$\frac{1}{2}(2 x)$$

Step-by-Step Solution

Answer

The expression simplifies to \(x\).

Step 1: Understand the Associative Property

The associative property states that the way in which numbers are grouped doesn't affect their sum or product. For multiplication, this means \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). We will use this to rearrange the expression \(\frac{1}{2}(2x)\).

Step 2: Apply the Associative Property

Apply the associative property to rewrite \(\left(\frac{1}{2}\right)(2x)\) in a different grouping. Original expressions are usually grouped as \(a(bc)\) equivalent to \((ab)c\). Thus, \(\left(\frac{1}{2} \cdot 2\right)x\).

Step 3: Simplify the Expression

Simplify \(\left(\frac{1}{2} \cdot 2\right)x\). Calculate \(\frac{1}{2} \cdot 2\). This product is \(1\), since \(\frac{1}{2} \times 2 = 1\). Thus, the expression simplifies to \(1 \cdot x\).

Step 4: Final Simplified Expression

Simplifying further, \(1 \cdot x\) is simply \(x\), as any number multiplied by 1 is the number itself. Therefore, the fully simplified expression is \(x\).

Key Concepts

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. This involves combining like terms, reducing fractions, and eliminating any unnecessary components of the expression. When we simplify, we're making the expression easier to understand and work with, without changing its value.

In the example given, \(\frac{1}{2}(2x)\), simplification means reducing it to its most basic terms, which is just \(x\). We achieve this by using properties of numbers, like the multiplication property, to reorganize and reduce terms.

This process typically involves three main steps:

• Applying properties like associative, commutative, and distributive to reorganize the expression.
• Performing arithmetic operations like addition, subtraction, multiplication, and division to combine numbers.
• Simplifying any resulting forms, such as fractions or coefficients, to ensure the expression is in its most compact form.

Algebraic Expressions

An algebraic expression is a mathematical statement that includes numbers, variables, and operational symbols. These combinations represent quantities and relationships between them. For instance, the expression \(2x\) includes a coefficient (2) and a variable (x).

Algebraic expressions are foundational because they allow us to model real-world situations and perform calculations that involve unknown values. Significant elements of algebraic expressions include:

• Variables: Symbols (often letters) that can represent different values.
• Coefficients: Numbers that multiply the variables.
• Constants: Stand-alone numbers in the expression.

By understanding how to work with these components, we can solve equations and simplify expressions, making algebra one of the key subjects for solving problems in math and science.

Multiplication Properties

Multiplication properties help us understand how numbers work together when multiplied. They include the associative, commutative, identity, and zero properties, each with unique characteristics for simplifying expressions and solving problems.

In our example, the associative property was key. The associative property allows for the rearrangement of grouped numbers without changing the product. This means \(a \cdot (b \cdot c)\) is the same as \((a \cdot b) \cdot c\).

Here are some important properties of multiplication:

• Commutative Property: Changing the order of numbers does not affect the product, \((a \cdot b = b \cdot a)\).
• Associative Property: The way numbers are grouped does not affect the product.
• Identity Property: Any number multiplied by one remains unchanged, \(a \cdot 1 = a\).
• Zero Property: Any number multiplied by zero equals zero, \(a \cdot 0 = 0\).

These properties make manipulation of algebraic expressions simpler and more intuitive.