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Perhaps, before taking into account each one of the exercises that can serve as an example of the correct way to solve any operation, that has as its purpose to determine what is the quotient of two powers, **that besides having rational bases, these coincide completely, t**he most recommendable thing is to revise briefly the definition of this operation, in order to understand each one of these procedures in its just mathematical context.

## Division of rational and equal base powers

However, we should start by remembering that in the mathematical field the name Rational Power is known as any operation involving a potentiation in which the base is constituted by a fraction, **that is, a mathematical expression used to represent non-integer or non-exact quantities,** and constituted by a numerator and a denominator.

In the same way, the mathematical discipline indicates that this type of operations must be solved elevating the fraction to the natural number that serves as exponent, developing then the abbreviated multiplication that it constitutes, and finding answer when multiplying the fraction that serves as base, **by itself, as many times as this exponent indicates.**

As for the Division of powers of rational and equal base **it will be an operation in which it will be looked for to determine which is the quotient that can be obtained when dividing two powers,** that have as base fractions, that coincide between themselves in each one of their elements, independently of the values of each one of the exponents to which these fractions are elevated.

## Steps to solve a division of powers of rational and equal base

Once an operation of this type has been proposed, Mathematics indicates that a series of steps must be followed to allow a correct solution, **which must also be followed in the following order:**

- The first step will consist of checking that each of the powers that participate in the division
**coincide fully, in reference to their numerator and denominator.** - In the second instance, in order to carry out the division, a single fraction must be assumed as power, and the values of the natural numbers that exercise as quotient must be subtracted.
- Having a rational power base, the potentiation operation will be solved, following the mathematical formula indicated for it, that is, raising each element of the fraction separately to the exponent indicated.
- Finally, if possible
**, the fraction obtained from the potentiation operation should be taken to its simplest expression.**

Consequently, **the way of solving any division that raises powers of rational and equal base can be expressed in the following way:**

## Examples of rational and equal power division

However, the best way to approach the study of this operation may be through the observation of some examples, which allow us to see in practice how each of these steps are applied. **The following are some exercises that demonstrate how to resolve rational and equal division of powers:**

## Example 1

**Resolve the next operation:**

In order to comply with the postulate of the exercise, the exponents of these powers of rational base must then be subtracted, **which will be possible because there is only one base:**

Once this is done, the potentiation operation must continue, **elevating each element of the fraction to the pertinent exponent:**

Since the fraction cannot be simplified, it is assumed to be the result of the operation.

## Example 2

**Solve the next division:**

Once it has been determined that this is a division of rational powers, **where both factors coincide in their elements, the exponents of each power must be subtracted:**

The potentiation operation will then be resolved, elevating each element to the cube, without forgetting that since it is an operation with a negative base but with an odd exponent, t**he result will always give an equally negative fraction:**

Since it cannot be further simplified, this fraction is taken as the simplest form, and the solution to the operation of division of rational and equal powers.

## Example 3

**Solve the next operation:**

In the same way, seeing that they are powers of rational and equal base,** the operation will be solved subtracting the exponents of each one:**

Once this power operation is achieved, the mathematical law should be remembered, **which says that any fraction elevated to an exponent equal to the unit will result in the fraction itself:**

When this fraction is reached, the simplification should be continued, taking the simplest form reached as the final solution of the operation:

## Example 4

**Resolve the next operation:**

Since the bases of these rational base powers are elevated to negative exponents, perhaps the best thing to do before proceeding with the solution of the operation is to convert these fractions to positive exponents, **which is done simply by inverting the terms of the fraction:**

Once this point has been reached, the pertinent procedure must **then be applied in any division of powers on a rational and equal base:**

Once this has been done,** each element of the fraction must then be elevated to the exponent obtained:**

Since the fraction can no longer be simplified, it is then considered as the final answer of the operation.

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October 25, 2019