Problem 15 - Beginning and Intermediate Algebra

Question

Use an associative property to complete each statement. See Example 2. \((a+b)+c=\)_____________

Step-by-Step Solution

Answer

\(a+(b+c)\)

Step 1: Understanding the Problem

We are given the expression \((a+b)+c\) and need to use the associative property to re-arrange the grouping of terms in addition.

Step 2: Review Associative Property of Addition

The associative property of addition states that for any real numbers \(x\), \(y\), and \(z\), the equation \((x+y)+z = x+(y+z)\) holds true.

Step 3: Apply the Associative Property

Applying the associative property, the expression \((a+b)+c\) can be rewritten as \(a+(b+c)\) by changing the grouping.

Step 4: Rewrite the Expression

The expression \((a+b)+c\) when rewritten using the associative property becomes \(a+(b+c)\).

Key Concepts

Properties of Addition

Addition isn't just putting numbers together; it follows specific rules known as the properties of addition. Understanding these properties helps simplify calculations and solve algebraic expressions more easily.

• Commutative Property: This property tells us that the order in which we add numbers doesn't matter. It means that for any numbers, say \(a\) and \(b\), the addition \(a + b = b + a\).
• Associative Property: This property is about grouping. It states that no matter how you group the numbers in addition, the result will be the same. For instance, if you have \((a + b) + c\), you can also write it as \(a + (b + c)\).
• Identity Property: This property tells us there is a special number, 0, which, when added to any number, doesn't change its value. That is \(a + 0 = a\).

These properties are like the rules of a game, helping us play with numbers without changing their value.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. They are a way to show relationships between different values.

For example, in the expression \(a + b + c\), \(a\), \(b\), and \(c\) are variables that can take different values. Expressions like these are essential because they allow us to represent real-world situations in a mathematical format.

Working with Algebraic Expressions

• Variables: These are symbols (like \(x\), \(y\), \(a\)) that stand in for numbers.
• Constants: These are fixed numbers within the expression, like 2 or 5.
• Operations: Addition, subtraction, multiplication, and division are the operations that interact with the variables and constants.

Manipulating algebraic expressions often involves using properties like the associative property of addition to simplify or rearrange them.

Real Numbers

Real numbers form one of the most essential concepts in mathematics. They include every kind of number that you can think of. They are what we usually use in daily calculations.

Types of Real Numbers

• Natural Numbers: These are the simple counting numbers like 1, 2, 3, and so on.
• Whole Numbers: Like natural numbers, but including zero (0, 1, 2, 3,...).
• Integers: These include all whole numbers and their negatives (-3, -2, -1, 0, 1, 2, 3,...).
• Rational Numbers: Numbers that can be written as a fraction of two integers, such as \(\frac{1}{2}\) or \(-3\frac{3}{4}\).
• Irrational Numbers: Numbers that can't be expressed as a simple fraction, such as \(\pi\) or \(\sqrt{2}\).

Real numbers are used in every aspect of mathematics, from simple addition to complex calculus, making them foundational for understanding algebraic expressions and the properties of addition.