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Chapter 2: Problem 26

Apply the associative property to expression, and then simplify the result. \((6 a+3)+14\)

Short Answer

Expert verified
The expression simplifies to \(6a + 17\) using the associative property.

Step by step solution

01

Identify the structure of the expression

The given expression is \((6a + 3) + 14\). Notice that it consists of two parts being added together: \((6a + 3)\) and \(14\). The associative property will help us rearrange these parts.
02

Apply the associative property

The associative property allows us to regroup elements in an addition operation. We can rewrite the expression as \(6a + (3 + 14)\). Notice that the parentheses have been shifted to include \(3 + 14\) instead.
03

Simplify inside the parentheses

Now, simplify the expression inside the parentheses: \(3 + 14 = 17\).
04

Write the simplified expression

After simplifying, the expression becomes \(6a + 17\). This is the final simplified form of the original expression after applying the associative property.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expression Simplification
Expression simplification is the process of making a mathematical expression as concise and straightforward as possible. In mathematics, it's important to simplify expressions to make calculations easier and to reveal the essence of the problem. To simplify an expression:
  • Combine like terms: Group similar terms together, such as all the constant numbers or all the terms involving variables like \(a\).
  • Use mathematical properties: Utilize properties such as associative, commutative, and distributive to rearrange and simplify parts of the expression.
  • Evaluate operations within parentheses first: This is often where simplification begins in accordance with the order of operations.
For instance, when simplifying \((6a + 3) + 14\), you use the associative property to regroup and simplify it to \(6a + 17\). By breaking it down, you make the problem much more manageable.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They don't have an equals sign, which distinguishes them from equations.Key components of algebraic expressions:
  • Variables: Symbols like \(a\) which represent unknown or changeable numbers.
  • Constants: Specific numbers on their own, such as \(3\) and \(14\).
  • Operators: Addition, subtraction, multiplication, and division symbols that combine different terms.
Algebraic expressions allow us to model real-world situations mathematically. They form the building blocks for equations and complex functions. In the exercise, \((6a + 3) + 14\), the expression is composed of a term with a variable \(6a\) and two constant terms \(3\) and \(14\), combined using an addition operator.
Prealgebra Concepts
Prealgebra introduces basic mathematical concepts that form the foundation for algebra. It focuses on understanding number operations, properties, and the relationships between numbers.Important prealgebra concepts include:
  • Number operations: Understanding addition, subtraction, multiplication, and division.
  • Mathematical properties: Learning how and why properties like associative, commutative, and distributive work.
  • Expressions and equations: Beginning to identify and manipulate these to solve basic problems.
In our example, applying the associative property to the expression \((6a + 3) + 14\) is a prealgebra concept that helps simplify the expression to \(6a + 17\). Understanding this property allows for flexibility in how we rearrange and tackle problems.

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Most popular questions from this chapter

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