Rational base powers and natural exponent

Perhaps the best thing, before delving into an explanation of the Rational and Natural Base Powers, is to briefly revise some definitions, which will allow us to understand this operation within its precise mathematical context.

Fundamental definitions

In this sense, it may also be appropriate to delimit this mathematical explanation to two specific notions: The Empowerment and the Fraction, since these are respectively the operation and the constituent expression of the mathematical procedure to be desired Study. Here are each of these definitions:

Power: definition and elements

In this way, one can begin by saying that the Mathematics has defined the Empoweror as an operation by means of which a certain number multiplies itself, as many times as a second element points out, in order to obtain the product of this procedure, a fact that makes the Power-up also understood by some authors as an abbreviated multiplication. This operation may be expressed mathematically as follows:

an = an . an2  . an

As for the elements that make up the operation called Powering, three of them will be distinguished, each of which will have its own definition and function, as can be seen below:

  • Base: it will consist of the number that will multiply by itself, as many times as indicated by the other number involved in the operation.
  • Exponent: on the other hand, also consisting of a number, the exponent will fulfill the function of telling the base how many times it must multiply on its own.
  • Power: Finally, the Power will be assumed as the result of the operation.


It will also be necessary to cast lights on the definition of Fraction, understood as one of the two expressions with which fractional numbers count, that is, that the fraction will be used to account for non-whole or non-exact amounts. Likewise, mathematical discipline indicates that fractions shall be constituted without exception by two elements, each of which has been explained as follows:

  • Numerator: First, the Numerator will occupy or make up the top of the fraction. Your mission will be to indicate how many parts of the whole have been taken, or represent the fraction.
  • Denominator: In the second instance, the Denominator will be the element that conforms to the bottom of the fraction, having as tare indicate in how many parts the whole is divided, of which the numerator points only a few parts.

Rational base powers and natural exponent

Bearing in mind each of these definitions, it is perhaps certainly much easier to address the concept of Rational-based powers and natural exponent, which can be basically explained as those empowering operations where the base is consists of a rational number or fraction, while the exponent is a natural number, that is, a positive integer. These expressions may be raised mathematically as follows:

Like any enhancer operation, the rational base and natural exponent powers will be solved by multiplying the base by itself, as many times as the exponent is pointed out. It is also necessary to remember that the multiplication of fractions is solved by multiplying the numerators by numerators, and denominators by denominators. Therefore, the solution of such an operation could be expressed in the following mathematical terms:

Example of how to solve a rational base power and natural exponent

However, perhaps the best way to complete an explanation of how such an operation should be solved, either through the exposure of an example, that allows concrete understanding of how each of the steps is applied. Here’s one of them:

Solve the following potentiation:

In order to solve this operation it will be necessary to multiply the fraction by itself three times, as indicated by the exponent that conforms to it:

Once the power result has been obtained, it will also be seen if this expression can be simplified:

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Rational base powers and natural exponent
Source: Education  
September 30, 2019

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