Problem 54 - College Algebra

Question

state the name of the property illustrated. $$ 7 \cdot(11 \cdot 8)=(11 \cdot 8) \cdot 7 $$

Step-by-Step Solution

Answer

The property being illustrated is the associative property of multiplication.

Step 1: Analyse the given expression

Given the expression \(7 \cdot(11 \cdot 8)=(11 \cdot 8) \cdot 7\), it is clear that the order of multiplication is not influencing the result.

Step 2: Identify the mathematical property

According to the property where the way you group the numbers when you multiply does not change the product, we identify the property as the associative property of multiplication.

Step 3: State the result

The property being illustrated in the given equation is the associative property of multiplication.

Key Concepts

Algebra and Multiplication

Algebra involves the manipulation of symbols and numbers according to set rules, one of which is the associative property of multiplication. This concept is foundationally important as it allows flexibility in how we group numbers in equations. To understand it better, consider the example '7 times the product of 11 and 8 equals the product of 11 and 8, times 7'. Algebraically, this demonstrates that regardless of how you group the numbers, the end product remains the same. This is a crucial aspect of algebra that simplifies complex problems, making them more approachable. When we apply the property correctly, it becomes a powerful tool in solving algebraic equations, allowing students to focus on simplifying problems without worrying about changing the outcome.

Mathematical Properties

Mathematical properties are rules that hold true for numbers and operations. One of these, the associative property of multiplication, tells us that the way in which factors are grouped does not affect the product.

For instance, the expression \(7 \cdot(11 \cdot 8)=(11 \cdot 8) \cdot 7\) demonstrates this property perfectly. It highlights that multiplication can be carried out in any order without changing the result. This property is applicable only to addition and multiplication operations and does not work for subtraction or division. Recognizing and correctly applying this property helps in simplifying mathematical expressions and is an integral part of understanding how different operations interact within the realm of mathematics.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which parts of a mathematical expression should be solved. However, because of the associative property of multiplication, when it comes to the multiplication part of this sequence, we have some flexibility.

The equation \(7 \cdot(11 \cdot 8)=(11 \cdot 8) \cdot 7\) is a perfect illustration that when we're only multiplying, we can do so in any order; the grouping of numbers doesn't matter. In contrast, for operations like subtraction and division, the sequence and grouping matter greatly. Understanding the order of operations, along with mathematical properties such as the associative property, allows students to accurately solve complex mathematical problems, ensuring they are following the correct process to reach the correct answer.