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Chapter 0: Problem 81

In Exercises \(75-84,\) state the name of the property illustrated. $$2(-8+6)=-16+12$$

Short Answer

Expert verified
The property illustrated in the exercise is the Distributive Property of Multiplication over Addition.

Step by step solution

01

Analyze the two sides of the equation

Looking at the exercise, we see that on the left side, \(-8\) and \(6\) are added together first, then the result is multiplied by \(2\). On the other side, \(2\) is individually multiplied by \(-8\) and \(6\).
02

Identify the mathematical property

The principle being used here is the Distributive Property of Multiplication over Addition. This property states that for all real numbers a, b, and c: a(b + c) = ab + ac. This matches the given equation, where \(a=2, b=-8, c=6\).

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Key Concepts

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

Multiplication
Multiplication is a fundamental arithmetic operation characterized by the repeated addition of a number. For example, if you multiply 2 by 3, you can think of it as adding 2 three times: 2 + 2 + 2 = 6. This process can be neatly summarized as 2 x 3 = 6. Here are some key points about multiplication:
  • Multiplication involves scaling one number by another.
  • The result of multiplying two numbers is called the product.
  • A number that is multiplied by another is referred to as a factor.
In the context of the Distributive Property seen in the exercise, multiplication occurs twice. The number 2 is multiplied by each component within the parentheses, demonstrating how multiplication can distribute over addition effectively. This showcases multiplication's flexibility and utility in combining numbers.
Addition
Addition is another basic arithmetic operation, focusing on combining numbers to find a total. It's perhaps the easiest operation to visualize: adding 3 to 2 simply means starting with 3 and counting up two more steps to reach 5.
Here are essential points about addition:
  • Addition is commutative, meaning a + b is the same as b + a.
  • It is associative: (a + b) + c equals a + (b + c).
  • The number zero is the identity element for addition, as a + 0 = a.
In our exercise, addition is seen within the parentheses (-8 + 6). First, the sum of these two numbers is calculated before multiplication. When applying the Distributive Property, multiplication is performed on each term, linking addition directly to multiply the results effectively.
Real Numbers
Real numbers include all the numbers you can think of, from positive to negative, whole to fractional, and everything in between, including irrational numbers like \( \pi \). Here's what you need to know about real numbers:
  • They can be represented on a number line.
  • Real numbers include rational numbers (fractions and integers) and irrational numbers.
  • They do not include imaginary numbers.
In the given exercise, real numbers are featured in the expression \(-8+6\) which represents a basic operation on real values. The Distributive Property of Multiplication over Addition is expressed with these real numbers, illustrating how operations are carried out with them. Real numbers are versatile and form the basis for most mathematical computations and properties discussed in mathematics.

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