Before moving forward on an explanation of the various Mathematical Properties, present in the powers of integers, it is likely that some definitions, which will allow us to understand each of these laws in their context, should be briefly reviewed, which will allow us to understand each of these laws in their context precise mathematician.
Fundamental definitions
It may also be prudent to delimit this theoretical review to two precise notions. The first, the definition of Whole Numbers, in order to take into account the nature of the numerical elements involved. Likewise, it will be of great importance to review the concept of Powers of integers, as this is the mathematical operation on the basis of which each of these mathematical properties happens. Here are each of these definitions:
Integers
In this way, the concept of Integers will be first addressed, which have been explained by Mathematics as the numerical elements through which the exact quantities are represented. Likewise, Integers are understood as the elements on which the numeric set of the same name is constituted, or also known as the Z set, and where these numbers are grouped as follows:
Positive integers: on the one hand, positive integers – a number that in turn constitute the Natural Numbers – will be placed to the right of the zero in the Number line. They shall extend from 1 to ∞, and through them it may account for accounting quantities, or count the elements of a set.
Negative integers: secondly, negative integers will be found, numbers that are considered the inverses of negative integers, which is why they will be located to the left of zero, in the number line, extending from -1 to ∞. These numbers will be used to express debts or misdemeanours of specific amounts, and must always be noted in the company of the minus sign.
Zero: Finally, zero is also considered an element of the Z set. However, it is not had as a number, but as the total absence of quantity. Consequently, this element will be neither positive nor negative, and will be considered inverse of itself.
Total number powers
In another order of ideas, it will be equally important to cast lights on the definition of Powers of integers, which can be explained broadly as the mathematical operation, occurring strictly in relation to whole numbers, where a first number of this type (which will be called base) chooses to multiply itself, as many times as it points to a second number (known as an exponent) for the purpose of knowing the product (which will receive the power name).
Integer power properties
With these definitions in mind,it is perhaps certainly much easier to address the explanation about each of the Mathematical Properties that take place in the Integer Powers operation, and can be listed as follows:
Integer powers with zero exponent
First, it will then be that whenever there is a power operation involving integers, and that power has an exponent equal to zero, regardless of the amount or sign of the base, the result will always be equal to one. This property may be expressed as follows:
a0 = 1
Integer powers with exponent one
On the other hand, Mathematics also points out that any potentiation operation that is given on the basis of integers, where the exponent is equal to 1, beyond the value or sign of the base, will always obtain as potency the equivalent number that has served as the basis. This property will be expressed mathematically as follows:
a1 = a
Product of integer powers of equal base
Thus, within the different mathematical properties that take place in the powers with integers you will find this that refers to the sustained situation among two powers of equal basis have chosen to multiply. In this case, Mathematics notes that the property orders that a single base is assumed, the values of its exponents are summed, and finally the base is raised to the total of the exponents. This situation may be explained as follows:
am . an = am+n
Equally base integer power ratio
Similarly, powers of equal base may decide to divide. In this case, the mathematical property indicates that a single basis is assumed, the value of its exponents subtracted, and finally raise the base to the difference of exponents, a situation that can be explained mathematically as follows:
am : an = am-n
Ownership of a power of integers
Another of the mathematical properties that can be found in relation to the powers of integers is the so-called Power of a Power, where this discipline then orders the exponents involved to multiply, and then raise the base to this obtaining the required power. This property will be expressed as follows:
(am)n = am.n
Distributive property in the multiplication of powers of integers and equal exponent
It may also result in two powers of different bases but of equal exponents choosing to multiply. In this case, the Distributive Property may be applied, allowing it to be resolved in two ways:
The first one will allow to solve the product of the base, and then raise this result to the common exponent:
- an . bn = (a.b)n
Secondly, each of the bases can be raised to the common exponent, and then multiplied by each of the powers:
(a. b)n = an . bn
Distributive property in the power division of integers of equal exponent
Also, distributive property may be available when facing the division of powers involving whole numbers, and have a single exponent. In this case, such transactions can also be solved in two ways, thanks to the distributive property:
In this way, it can be solved by calculating in base quotient, and then by elevating this result to the common exponent:
am: bm= (a: b)m
Another option to solve this type of operations will be one where each of the bases is raised to the common exponent, and then proceed to split the different quotients.
(a: b)m = am : bm
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