Problem 13 - Beginning and Intermediate Algebra

Question

Use an associative property to complete each statement. See Example 2. \(4 \cdot(a b)=\)_______

Step-by-Step Solution

Answer

\((4a)b\)

Step 1: Identify the Associative Property

The associative property of multiplication states that for any numbers \(a, b,\) and \(c\), the equation \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) holds true. This means that the grouping of the numbers does not affect the product.

Step 2: Apply the Property to the Given Expression

In the given expression, \(4 \cdot (a \cdot b)\), use the associative property to regroup the multiplication. Adjust the expression to \((4 \cdot a) \cdot b\).

Step 3: Write the Completed Statement

Using the associative property, the original expression \(4 \cdot (a \cdot b)\) is equivalent to \((4 \cdot a) \cdot b\). The completed statement is as follows: \(4 \cdot (a b) = (4a)b\).

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that allows us to calculate the product of two numbers. It involves adding a number to itself a certain number of times. If you think about multiplying 4 by 3, it’s the same as adding 4 together three times:

• \(4 + 4 + 4 = 12\)

In algebra, multiplication becomes a building block for more complex calculations. When we multiply numbers, we often encounter properties that make our calculations easier. One such property is the associative property, which states that when multiplying three or more numbers, the way in which the numbers are grouped does not change the product. This means for any numbers \(a, b,\) and \(c\), the equation \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) holds true. This idea can be applied to simplify expressions and solve problems more efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. Variables, represented by symbols like \(a, b,\) or \(x,\) stand in for unknown values or quantities that can change. They play a vital role in forming expressions that represent real-world situations. These expressions can take a simpler appearance, like \(4a\), or be more complex, encompassing several terms and operations. The importance of algebraic expressions lies in their ability to generalize arithmetic operations and solve problems beyond the straightforward calculations of numbers alone. When using the associative property in the context of algebraic expressions, we reinforce the flexibility of grouping the terms differently without disrupting the overall relationship represented by the expression.This makes algebra an essential tool in both theoretical scenarios and practical applications.

Properties of Operations

Understanding the properties of operations like the associative property is crucial in mathematics. These properties provide a framework for how numbers and operations interact, helping simplify complex problems and support mathematical reasoning. For multiplication, three key properties stand out:

• Commutative Property: This states that the order in which two numbers are multiplied does not affect the product, i.e., \(a \cdot b = b \cdot a\).
• Associative Property: The focus of our exercise, this property tells us that how numbers are grouped during multiplication does not alter the result, so \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Distributive Property: This property allows us to multiply a number by a sum by distributing the multiplication over each addend, expressed as \(a \cdot (b + c) = a \cdot b + a \cdot c\).

Using these properties, we can reorganize, simplify, and solve expressions efficiently. Mastery of these concepts can lead to a deeper understanding of algebraic principles, supporting the development of more advanced mathematical skills.