Problem 115 - Intermediate Algebra

Question

Use variables to state each property of real numbers. a. Commutative property of addition b. Associative property of multiplication c. Distributive property of multiplication over addition

Step-by-Step Solution

Answer

a. \( a + b = b + a \) b. \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) c. \( a \cdot (b + c) = a \cdot b + a \cdot c \)

Step 1: State the Commutative Property of Addition

The commutative property of addition states that changing the order of the addends does not change the sum. Algebraically, this can be expressed as: \( a + b = b + a \).

Step 2: State the Associative Property of Multiplication

The associative property of multiplication describes that the way in which factors are grouped in a multiplication problem does not affect the product. Algebraically, this property is expressed as: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).

Step 3: State the Distributive Property of Multiplication Over Addition

The distributive property of multiplication over addition states that a number can be multiplied by a sum by distributing the multiplication over each addend individually and then adding the products. Algebraically, this is represented as: \( a \cdot (b + c) = a \cdot b + a \cdot c \).

Key Concepts

Commutative Property

The commutative property is a fundamental principle used in mathematics, particularly in relation to addition and multiplication. This property tells us that we can swap numbers around without changing their result. When it comes to addition, the commutative property is evident when rearranging numbers. For example, if you have two numbers, 3 and 5, the order in which you add them does not matter:

• \( 3 + 5 = 8 \)
• \( 5 + 3 = 8 \)

Whether you add 3 to 5 or 5 to 3, the sum remains the same. This simple observation allows more flexibility in calculations and provides a foundational block upon which more complex mathematical concepts are built.

Associative Property

The associative property refers to the way numbers are grouped in an operation and how this grouping does not change the result of their addition or multiplication. An easy way to understand this with multiplication is to group numbers in any way, and the product remains the same:

• \( (2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4) \)
• In both cases, the result is 24.

This concept simplifies complex calculations by allowing you to regroup terms in a way that might make mental math quicker or more intuitive.

Distributive Property

The distributive property is a bridge between addition and multiplication. It lets us multiply a sum by breaking it down into individual multiplications and then adding the results. This property can be illustrated as follows:

• \( 2 \cdot (3 + 4) = 2 \cdot 3 + 2 \cdot 4 \)
• The sum 3 + 4 is multiplied by 2, both individually and collectively, yielding the same product of 14.

By breaking complex expressions into simpler parts, the distributive property makes problem-solving more manageable, especially when dealing with algebraic expressions or larger numbers.

Addition

Addition is one of the most basic operations in arithmetic. It involves combining two numbers to form a new total. Simple examples like \( 4 + 7 = 11 \) show how straightforward addition is. It is extensively used in daily life, from computing totals at a store to figuring out how many apples remain if you start with a dozen and give away some.
Understanding addition sets the groundwork for learning other mathematical concepts, as it is used frequently in algebra, calculus, and various other fields of math.

Multiplication

Multiplication is an extension of addition. Instead of adding a number several times, multiplication allows for a quicker computation. For instance, adding 5 three times (\(5 + 5 + 5\)) can be swiftly expressed as \(3 \cdot 5 = 15\).
This operation simplifies calculations and is a core skill in math education.

• Multiplication tables are often used to help memorize the basic products.
• They provide answers to simple multiplication problems and are essential for handling more advanced mathematical tasks.

Multiplication serves as the basis for concepts such as exponents, where numbers are raised to powers.