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Chapter 1: Problem 10

State the property of multiplication depicted by the given identity. $$ 8 \cdot 7=7 \cdot 8 $$

Short Answer

Expert verified
The identity exemplifies the commutative property of multiplication.

Step by step solution

01

Identify the multiplication statement

We are provided with the multiplication identity \( 8 \cdot 7 = 7 \cdot 8 \). We need to state which property of multiplication this identity represents.
02

Understand the commutative property

The commutative property of multiplication states that changing the order of factors does not change the product. In other words, for any two numbers \( a \) and \( b \), \( a \cdot b = b \cdot a \).
03

Apply the commutative property to the identity

In the given identity \( 8 \cdot 7 = 7 \cdot 8 \), we can see that the order of multiplication does not affect the product. Hence, this equation exemplifies the commutative property of multiplication.

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Key Concepts

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

Multiplication
Multiplication is one of the basic operations in mathematics, often used to describe repeated addition. For example, multiplying 3 by 4 (\(3 \cdot 4\)) means you are adding 3 together, 4 times, which results in 12. There are some key points to remember about multiplication:
  • It is binary, meaning it operates with two numbers known as factors.
  • The result of a multiplication operation is called the product.
Multiplication is also fundamental in various areas of mathematics and everyday life, such as computing the area of a rectangle or scaling recipes in cooking. The operation simplifies the process of combining equal groups and serves as the basis for more advanced concepts in mathematics.
Mathematical Properties
Mathematical properties are rules that apply to mathematical operations, ensuring consistency and reliability. The commutative property is one such rule that plays a crucial role, especially in arithmetic operations like multiplication and addition.The commutative property of multiplication states that changing the order of factors does not affect the product. For instance:
  • \(a \cdot b = b \cdot a\)
  • Using numbers, this means \(8 \cdot 7 = 7 \cdot 8\).
This property simplifies calculations and allows flexibility. You can rearrange numbers without worrying about changing the result, which is pretty handy when solving complex problems or when programming computers to perform multiplication.
Order of Operations
When performing mathematical calculations, the order in which operations are executed is crucial for obtaining the correct result. This concept is known as the 'order of operations,' and it's a set of rules that dictate which operations to perform first. A common acronym to remember this order is PEMDAS:
  • P: Parentheses first
  • E: Exponents or powers (i.e., squares, cubes)
  • M/D: Multiplication and Division from left to right
  • A/S: Addition and Subtraction from left to right
Understanding the order of operations ensures clarity and precision in mathematical solutions. By performing multiplication (and division) before addition (and subtraction), mathematicians maintain consistency across various calculations, preserving the integrity and predictability of results.

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