# Examples of how to solve products of rational powers of equal base

Perhaps the most convenient thing, prior to exposing some exercises that can serve as an example to the correct way in which any operation that raises the need to find the product of powers of rational base, that fully coincide as to their bases, must be solved, is to review briefly some theoretical matters that Mathematics has pointed out regarding this procedure.

## Power products of rational and equal bases

Thus, one should begin by remembering that Mathematics has explained fractions as a type of mathematical expression, which fulfills the task of accounting for fractional numbers, or what is equal, for non-exact or non-integer quantities. Likewise, the mathematical discipline conceives fractions composed of two elements: numerator, which occupies the upper part, expressing how many parts of the whole have been taken; and denominator, whose function is to occupy the lower part of the fraction, expressing how many parts the whole is divided into.

As for the rational base Power, this discipline has indicated that it can be considered an operation of potentiation in which the number that serves as base is composed by a rational number or fraction, and that it will have to be solved elevating each one of the elements of the fraction to the indicated exponent. Consequently, the product of powers of rational and equal bases will consist in the multiplication that is made between two or more fractions, which present as base fractions, which coincide with each other, in all their elements, beyond the particular value of each one of their exponents.

## Steps to solve rational and equal base powers

Likewise, Mathematics has indicated a series of necessary steps at the time of giving solution to this type of operations, which will have to be followed in the following order:

1.  Once an operation involving the multiplication of rational base powers has been given, each of the elements of the fractions constituting the factors of said multiplication must be reviewed in order to corroborate that they are rational base powers in which there are equal bases.
2. With this certainty, a single fraction will then be assumed as a base, and at the same time the exponents that both fractions had will be added.
3. Once a rational base power has been obtained, the operation continues, elevating each element of the fraction to the exponent obtained from the sum.
4. If possible, the fraction is simplified until the simplest expression of it is obtained.

On the other hand, the operation to be applied in this type of cases can then be expressed in the following way:

## Examples of how to solve products with rational and equal power bases

However, it is possible that the best way to study the correct way in which this type of operations should be solved is through the exposition of some examples, where it can be seen in a practical way how each one of the steps, considered by Mathematics, should be applied.

## Example 1

Solve the following rationally based power product:

When we begin to solve this operation, we will see that the factors are powers that, in addition to having a rational base, coincide with each other, so we must then proceed to add their exponents:

At this point, the operation will continue, elevating each of the elements of the rational base to the exponent obtained, and then -if possible- the fraction must be simplified:

## Example 2

Resolve the next operation:

In this case, there are two factors, which fully coincide in terms of their elements, even though only one of them is clearly elevated to an exponent, which of course doesn`t mean that the second factor is not. In fact, this could be interpreted as an operation of multiplication of powers of rational and equal base, where one of the powers of rational base is elevated to unity. Therefore, it will be solved by adding the exponents of each base and assuming only one:

At this point, a solution will be given to the operation as should be done in cases of rational base powers, elevating each of the elements to the indicated exponent:

## Example 3

Solve the following multiplication:

Even if it is a multiplication between more than two factors, being powers of rational base, which coincide as to their bases, it must be solved equally assuming a single base, and adding its exponents, even those that aren´t expressed, and that will always be taken as equivalent to unity:

Once this rational base power has been obtained, the operation will continue by elevating each element of the base to the indicated exponent, and if possible the fraction obtained will be simplified:

## Example 4

Resolve the next operation:

In this case, a multiplication of powers of rational and equal base is presented, which are elevated to negative exponents. In such circumstance, the most recommendable thing will be to convert these powers first in operations with positive exponents, for which simply the elements of each fraction will be inverted:

At this point, it will be possible to continue with the solution of the exercise, determining that it is a product of rational and equal bases, so it will be possible to assume a single base and add its exponents:

Once this has been done, each element will be elevated to the corresponding exponent, and if possible, the operation obtained should be simplified:

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Examples of how to solve products of rational powers of equal base
Source: Education
October 25, 2019