Problem 25 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((12 a+2)+19\)

Step-by-Step Solution

Answer

The simplified expression is \(12a + 21\) after applying the associative property.

Step 1: Identify the parts of the expression

The given expression is \((12a + 2) + 19\). This is a sum of three values: \(12a\), \(2\), and \(19\).

Step 2: Apply the associative property

The associative property states that we can change the grouping of the numbers being added. Thus, instead of \((12a + 2) + 19\), we can write the expression as \(12a + (2 + 19)\).

Step 3: Simplify inside the parentheses

Next, simplify the expression inside the parentheses \((2 + 19)\). Calculate \(2 + 19 = 21\).

Step 4: Rewrite the expression

Now, substitute the simplified result back into the grouped expression: \(12a + 21\). This is the simplified form after applying the associative property.

Key Concepts

Expression Simplification

When tackling algebraic expressions, simplifying complex expressions into simpler forms can make calculations much more manageable. The original expression given here is \((12a + 2) + 19\). Simplification involves removing unnecessary elements or combining like terms to reduce confusion and error potential in later calculations. By simplifying expressions:

• We make them easier to handle and interpret.
• It helps identify common factors or terms that can be further reduced.

In the given expression, it’s crucial to understand which terms can be added or multiplied together. By using appropriate properties like the associative property in this expression, we can rearrange and simplify to achieve a cleaner result.

Basic Algebra

Basic algebra forms the foundation of more advanced mathematical studies. It uses symbols and letters to represent numbers, allowing the formulation and solving of equations. In basic algebra, operations such as addition, subtraction, multiplication, and division are performed on variables and constants following specific rules. Understanding these rules is vital when working with expressions like \((12a + 2) + 19\). In basic algebra:

• Each term in an expression represents a single piece of information or calculation.
• Variables (like \(a\)) are used to stand in for unknown values.

Using basic algebraic rules, we enable more in-depth exploration and manipulation of complex mathematical problems.

Mathematical Properties

Mathematical properties are rules that provide structure to operations across numbers, helping define how we approach problems. For this problem, the critical property is the associative property of addition. This property states that how numbers are grouped in an addition operation does not impact the final result. This property allows us to change the grouping in the expression \((12a + 2) + 19\) to \(12a + (2 + 19)\):

• It emphasizes that the sum is independent of the grouping.
• Allows us to simplify calculations by combining certain terms first.

Understanding these properties empowers students to approach algebraic expressions with confidence and flexibility, knowing that they have systematic rules to guide them to a solution.