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Chapter 2: Problem 1

In the multiplication of 8 脳 7 = 56, what are the names given to the 8 and 7 and the 56?

Short Answer

Expert verified
8 and 7 are multiplicands (or factors), and 56 is the product.

Step by step solution

01

Identifying the Multiplicands

In the mathematics of multiplication, the numbers being multiplied are called 'factors.' Specifically, in the equation \(8 \times 7 = 56\), the numbers 8 and 7 are referred to as 'multiplicands.' Each multiplicand can also be individually referred to as a factor.
02

Recognizing the Product

The result of multiplying two or more numbers is called the 'product.' In the equation \(8 \times 7 = 56\), the number 56 is known as the product. It is the outcome of multiplying the multiplicands 8 and 7 together.

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Key Concepts

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

Understanding Multiplication
Multiplication is one of the four basic arithmetic operations and is essentially repeated addition. It involves combining equal groups of numbers to find a total. For example, when multiplying 8 by 7, you're essentially adding the number 8 to itself 7 times or vice versa.
  • Notation: In mathematics, multiplication can be indicated by the symbol \(\times\), a dot \(\cdot\), or simply by placing numbers next to each other (like in algebra).
  • Commutative Property: One of the fundamental properties of multiplication is the commutative property, which states that the order of the numbers in multiplication doesn't matter. So, \(8 \times 7\) is the same as \(7 \times 8\).
  • Association: The associative property allows you to change the grouping of numbers when multiplying. For instance, when dealing with more than two numbers, \((2 \times 3) \times 4\) gives the same result as \(2 \times (3 \times 4)\).
This is why multiplication is so versatile and crucial for solving more complex mathematical problems.
What Are Factors?
In multiplication, the numbers that are multiplied together are called factors. For instance, in the problem \(8 \times 7 = 56\), 8 and 7 are the factors. These values are sometimes also referred to as multiplicands, but "factors" is the more common term.
  • Definition: Factors are numbers that divide another number completely, without leaving a remainder. In the context of multiplication, they are what you multiply to achieve a product.
  • Prime vs Composite Factors: Factors can be prime (only divisible by 1 and itself) or composite (having multiple divisors), like 8, which can be broken down into 2, 2, and 2.
  • Real-world Use: Recognizing factors is useful in various applications, such as simplifying fractions or finding the greatest common divisors.
Understanding factors is essential in simplifying and solving mathematical problems efficiently.
The Concept of a Product
The product is the result you get from multiplying factors together. It is an integral part of multiplication, representing the total quantity when equal groups of numbers are combined. For instance, in \(8 \times 7 = 56\), the number 56 is the product.
  • Definition: A product is the final result or outcome of a multiplication operation. It showcases the cumulative value generated by multiplying the factors.
  • Importance: Products are essential in fields such as finance, science, and engineering where calculations of totals or measurements are routine.
  • Real-life Example: If you were buying 8 packs of pencils, each containing 7 pencils, understanding the product helps you determine that you will have 56 pencils in total.
Grasping the concept of a product is key to employing multiplication in everyday situations effectively.

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