Rational and exponent-based powers 1

Perhaps best, before addressing an explanation of mathematical property present in any rational-based power that is elevated to an exponent constituted by the unit, is to briefly revise some definitions, which will allow us to understand this law mathematical in its precise context.


Fundamental definitions

In this order of ideas, it may also be appropriate to delimit this theoretical review to three specific notions: Power-up, Fractions and Rational-Based Powers, as these are expressions and operations, directly related to property math that occurs when a fraction is raised to an exponent equal to 1. Here’s each one:

Empowerment

In this way, it will begin by saying that the Mathematics has defined the Empoweror as an operation, by which it seeks to determine the product that is obtained by multiplying a specific number by itself, as many times as indicated by a second element most sources choose to explain the Potentiation as well as an abbreviated multiplication. This operation can be expressed mathematically as follows:

an = an . an2  . an

Likewise, mathematical discipline has indicated that the Power-Up can be considered as an operation consisting of three elements, each of which has been explained as shown below:

  • Base: First, the base will be the numeric element, which must multiply itself as many times as the second number involved in the operation indicates.
  • Exponent: for its part, the exponent will be the second element of the empowering, having the task of indicating how many times the number that serves as the basis must be multiplied by itself.
  • Power: Finally, the Power will be interpreted as the final result of the operation, that is, the product resulting from the multiplication of the base by itself, as many times as the exponent has pointed out to it.

Fractions

As for the definition of fractions, Mathematics has indicated that these can be considered as a mathematical expression, by means of which rational or fractional numbers are counted, that is, that fractions serve to represent uncertain or unaccurate amounts. Thus, this discipline indicates that these expressions are composed of two elements:

  • Numerator: which will be understood as the number that makes up the top of the fraction, and whose purpose is to indicate how many parts of the whole represents the fraction:
  • Denominator: on the other hand, the denominator will be understood as the numerical element that occupies the bottom of the fraction, having the mission of indicating in how many parts the whole has been divided.

Rational-based powers

Finally, it will also be important to take a moment to reflect on the concept of Rational Powers, which will be understood as those power-up operations, which is based on a rational number, that is, a fraction, which is found raised to a natural number.

As in powers involving integers, in those rational-based powers the solution will find itself multiplying the base by itself, as many times as indicated by the exponent, which will be represented mathematically as follows:

Rational-based powers raised to 1

With these definitions in mind, it is perhaps certainly easier to address an explanation of the mathematical law present in every potential that, based on a rational number, has an exponent equal to unity, and which, as explained by the Mathematics should always result in the very fraction that has served as the basis.

Consequently, it could then be said that the unit in this case serves as a neutral element as well, since it does not produce any change in the number with which it shares operation. For its part, this property may be expressed mathematically in the following terms:

Examples of rational powers raised to 1

However, it is likely that the most efficient way to complete an explanation of the law that says about any rational-based power where an exponent equal to 1 can be seen is through the presentation of some examples that allow to see in a practical way how use this property without exception, always resulting in the operation being the base fraction itself. Here are some exercises:

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Rational and exponent-based powers 1
Source: Education  
September 30, 2019


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