Division of rational powers of equal base

Perhaps most convenient, before delving into an explanation of how any exercise that raises a division of fractional powers where it is determined that the factors are determined to be equal powers, be it be review some definitions, necessary to understand this operation in its just mathematical context.

Fundamental definitions

In this sense, it may also be prudent to delimit this theoretical review to three specific concepts: Power- Empowerment, Fractions and Rational-based Powers, as these are the operations and expressions directly related to the procedure splitting fractional powers of equal base. Here are each of these definitions:


It will therefore begin to say that the Mathematics has defined Potentiation as an operation aimed at determining the product of multiplying a numerical element on its own, as many times as the second number involved in the therefore, the Power-Up is also pointed out by the majority of authors as an abbreviated multiplication, a procedure which may be expressed mathematically as follows:

an = an . an2  . an

Likewise, the mathematical discipline has pointed to the Empoweror as an operation, which is composed of three elements, each of which has the following definition:

Base: this will be the number to be multiplied by itself, as many times as the second number with which the operation is formed.

Exponent: as for the Exponent, it will be made up of a numerical element, and it will have the responsibility to tell the base how many times it should multiply by it.

Power: will be interpreted as the final product of the Powering operation.


Likewise, it will be necessary to pause for a moment in the notion of Fractions, which have been broadly described by the different sources as a type of mathematical expression, by which fractional or non-whole amounts are represented. Thus, Mathematics indicates that Fractions will be composed of two elements:

  • Numerator: understood as the element that constitutes the top of the fraction, and whose main mission will be to indicate how many parts of the whole have been taken.
  • Denominator: On the other hand, the Denominator will be the element that will occupy the bottom of the expression. Your task will be to indicate how many parts the whole or unity has been divided into, of which the fraction expresses only some of them.

Rational-based powers

Finally, it will also be relevant to cast lights on the definition of Rational Base Powers, which have been described by Mathematics as a type of Power-up operation, where a fraction can be found as the basis. As with whole-base powers, the solution to this operation will be multiplying the base fraction as many times as the exponent points, which can be expressed mathematically as follows:

However, most authors prefer to apply in this type of operations the mathematical procedure understood as a general formula of fractional powers, which dictates that the best in this type of operations will be to elevate each element of the expression to the exponent that has been pointed out, a method which in turn will have the following expression:

Fraction Power Division with equal base

With these definitions in mind, it is perhaps certainly much easier to address an explanation of how any division operation that arises between fractional powers, where the same base is terminated, should be resolved.

In this sense, the various sources point out that there is a Law or Mathematical Property which mandates that whenever an operation with this approach is addressed, it will be decided to assume a single basis and subtract its exponents, as would be the case if the basis were made up of whole numbers. This mathematical property can be expressed as follows:


However, perhaps the most efficient way to complete an explanation of the property that is established in the case of division of fractional powers of equal base, is through the exposure of a specific example, which allows us to see in practice how each of  the steps inherent in solving such operations, as can be seen in the exercise below:

Resolve the following operation:

When you start to solve this operation, you will see that both factors are powers of equal base, so considering equally that they are divided, it will then be necessary to assume a single base, and subtract the exponents:

The result is equal to a fractional power raised to an exponent equal to 1. In this case, it is proceeded according to the mathematical law which states that whenever it is facing a power of fraction high to the unit, the product will be equal to the same fraction:

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Division of rational powers of equal base
Source: Education  
October 17, 2019

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