# Distributive property of integer powers with respect to multiplication

Before moving forward on an explanation of the mathematical law, known as Distributive Property in the Power of Integers with respect to multiplication, it may be necessary to revise some definitions, which will allow you to understand this property in its context precise mathematician. ## Fundamental definitions

As a result, it may also be necessary to focus this theoretical review on two specific notions: first, the definition of Whole Numbers should be taken into account, since these constitute the numerical elements on the basis of which the operation is given which gives rise to this property, which in turn receives the name Of Empowering of Whole Numbers, and the concept of which must also be revised. Here’s each one:

## Integers

Therefore, it will begin by saying that Mathematics has agreed to define whole numbers as those numerical elements representing whole and exact quantities, and which in turn are the elements on which the numerical set is constituted namesake, also called a Z-set.

Thus, Mathematics has indicated that within this numerical collection, integers are grouped into two subsets and one element, which have been explained by this discipline as follows:

Positive integers: These numbers will be grouped into the set of natural numbers, which is part of Z. They will be characterized by being to the right of zero in the number line, as well as by extending from 1 to infinity. Membership in the whole number will make this collection useful when counting the elements of a grouping, assigning them an order or position, or even expressing an accounting quantity.

Negative integers: Similarly, within the Z set you will find the subset of the negative integers, which will be made up of all the numbers from -1 to -, which are in the number line to the left of zero. Through these numbers, it is possible to account for the lack or debt of a specific amount.

Zero: Finally, zero will also be considered as a constituent element of the Z set. However, Mathematics does not consider this element as a number, but as the total absence of quantity. So whenever you want to express this mathematical notion you must make use of zero. Likewise, because it is not considered a number, it will not be taken as positive or negative, while assuming itself as an inverse of itself.

Empowering whole numbers

On the other hand, it will also be of great importance to throw lights on the concept that Mathematics gives with regard to the Empowering of whole numbers, which is then understood as a mathematical operation, constituted exclusively and obligatory by numbers integers, where one of these numbers must multiply itself as many times as it points to a second number, also integer, resulting in a product. Some authors prefer to define the Empowering of integers as an abbreviated multiplication of these numbers.

With regard to the elements of this operation, Mathematics also notes that the whole number that chooses to find a product multiplying by itself will be named the base. For its part, the number that indicates to this base how many times it should be performed this multiplication will be known as exponent, while the final product will be called power.

## Distributive property in integer powers relative to multiplication

Having reviewed these definitions, it may then be a little easier to cover the property pointed to by Mathematics in reference to the Distributive Property that can take place in any operation of multiplication of powers of integers.

In reference to this, it should be noted that this discipline warns that whenever there is a multiplication of whole numbers that is elevated to the same exponent, thanks to the Distributive Law there will be two possible ways of finding a solution to this Operation:

1.- In the first instance, it is possible to decide the product of the bases, and then raise it to the exponent to which the two numbers were raised. An example of this would be as follows:

(2 . 8)2 =  162

162 = 256

2.- However, you will also have the option to proceed to raise each of the numbers to the exponent that corresponds to it, and then multiply the powers of each of them, as will be seen below in the following example:

(2 . 8)2 = 22 . 82

22 . 82 = 4 . 64

4 . 64 =  256

As can be seen, thanks to the Distributive Property, each of the methods chosen will yield equal results.

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Distributive property of integer powers with respect to multiplication
Source: Education
September 21, 2019

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