Perhaps best, before moving forward with an explanation of the correct way to approach an operation that raises the elevation to an exponent of a rational and natural exponent power, is to take into account certain definitions, which will allow us to understand this operation in its just mathematical concept.
In this regard, it may also be most relevant to delimit this theoretical review to three specific notions: Potentiation, Fractions and Powers of Rationals, Operations and Expressions which are directly related to terms and procedures, inherent in determining the power of a fractional power. Here are each of these concepts:
In this way, it will begin by saying that the Power-up has been explained by the different mathematical sources as an abbreviated multiplication operation, where it is then sought to determine what is the product of multiplying a specific number by itself, as many times as a second numerical element points out, which can be expressed as follows:
an = an1 . an2 . an3 …
Thus, the various authors choose to describe the Power-up as an operation composed of three elements, explained in turn as can be seen below:
- Base: First, the Base will be understood as the number to multiply by itself, as many times as the second numerical element that participates in the operation points out.
- Exponent: Likewise, the Exponent will be understood as the number that tells the base how many times it should multiply by itself.
- Power: Finally, the Power is interpreted as the product of the multiplication of the base by itself as many times as the exponent points to it. That is, it is the end result of the operation.
In another order of ideas, it will also be prudent to address the definition of Fractions, which have been understood as a mathematical expression by which fractional, rational or non-exact amounts are realized. Mathematics also explains fractions as an expression composed of two elements:
- Numerator: on the one hand, the Numerator will constitute the top of the fraction, having as its mission to indicate how many parts have been taken altogether or are represented by this expression.
- Denominator: In the second instance, the Denominator will be the number that occupies the bottom of the fraction, and whose main task will be to indicate how many parts the whole is divided into.
Finally, it will be equally prudent to bring to chapter the concept of Rational-based Powers, which will be understood as those Power-Up operations, where bases of fractions can be found.
According to what the different authors point out this type of operations should be solved applying what in Mathematics is known as the general formula for fractional powers, and which will be understood on their part as a procedure by which it rises each element of the fraction to the exponent that it offers in an original way, which could be expressed in this way:
Power of rational base power
Once each of these definitions has been revised, it is perhaps certainly much easier to approach an explanation of the correct way forward whenever an operation is raised where a rational base power, made up of its base and its exponent, be elevated in turn to a second exponent. In such cases, as dictated by the inherent mathematical property, the correct thing to do will be to make the exponents, both the base and power exponents, multiply. This operation shall be expressed mathematically as follows:
However, perhaps the most efficient way to complete an explanation of how any potentiation operation that is elevated to an exponent should be solved, either through the exposure of a particular example, in which they can see how it is applied an each of the steps that indicates the mathematical property that exists to respect. Here’s the following exercise:
Solve the rational base power offered below:
Since the possibility of simplifying the operation is not considered, the fraction obtained as the final result of the operation is then taken.
October 17, 2019