Problem 22 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((8 x+3)+10\)

Step-by-Step Solution

Answer

The simplified expression is \( 8x + 13 \).

Step 1: Recognize the Initial Expression

The initial expression given is \( (8x + 3) + 10 \). We need to apply the associative property to this expression.

Step 2: Apply the Associative Property

The associative property states that when adding three or more numbers, the way in which the numbers are grouped does not change their sum. We apply this property by regrouping the expression: \( 8x + (3 + 10) \).

Step 3: Simplify Inside the Parentheses

Add the numbers inside the parentheses to simplify: \( 3 + 10 = 13 \). Thus, the expression becomes \( 8x + 13 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, enabling the representation of numbers and operations in a compact form. An algebraic expression consists of numbers, variables (like \( x \)), and operations such as addition and multiplication.

For instance, in the expression \( (8x + 3) + 10 \), there are:

• A variable term: \( 8x \), where \( 8 \) is the coefficient multiplying the variable \( x \).
• Constant terms: These are numbers without variables, such as 3 and 10.
• Operations: Addition is used to combine terms.

Every algebraic expression has a structure that can be manipulated using mathematical properties, such as the associative property. When you become familiar with these expressions, simplifying them using properties becomes straightforward.

The Role of Mathematical Properties

Mathematical properties are rules that help us work with numbers and expressions more efficiently. One such property is the associative property, which affects how we group numbers in addition or multiplication.

Specifically, in addition, the associative property tells us: \((a + b) + c = a + (b + c)\). This means the sum remains the same regardless of how the numbers are grouped.

• This property is particularly useful when dealing with complex algebraic expressions, as it allows us to reorganize or simplify them without changing their value.
• When applied to our original expression \( (8x + 3) + 10 \), the associative property allows us to rearrange it to \( 8x + (3 + 10) \).

Other properties like the commutative property and distributive property also support the manipulation and simplification of expressions, making mathematics more flexible and intuitive.

Simplification in Algebra

Simplifying expressions is a crucial skill in algebra, ensuring that expressions are in their simplest form. By simplifying, we make equations easier to understand and solve.

In our context, after applying the associative property to \((8x + 3) + 10\) to obtain \(8x + (3 + 10)\), the next step is simplification.

• This involves calculating the operation inside the parentheses: \(3 + 10 = 13\).
• Once simplified, the expression becomes \(8x + 13\), which is a cleaner, more understandable form.

Simplification often involves combining like terms, reducing expressions, and ensuring all operations are completed. Mastery of these techniques can significantly enhance problem-solving skills and make working with algebra more efficient.