Examples of how to solve a power of rational base and negative exponent

It is probable that, before approaching some exercises that can serve as example to the form in which all power of rational base and negative exponent must be solved, it will be convenient to revise of brief form the own definition of this operation, in order to understand each one of the mathematical procedures in its just context.


Powers of rational base and negative exponent

However, it may also be convenient to take a moment to remember that Mathematics calls as rational base powers any potentiation operation based on a fraction. It also points out that this fraction will always be elevated to a natural number. The way to give solution to this type of approaches will be through the elevation of each element of the fraction to the indicated exponent.

However, not always the natural number that serves as base is constituted by a positive number, being able to find sometimes a negative exponent, which if it is added to the fact that the base of the power is a fraction, then the operation of power of rational base and negative exponent will be had.

Steps to solve a power of rational base and negative exponent

In this way, Mathematics has also indicated which are the steps that must be followed at the time of giving solution to this type of operations, constituting a method that will have to be followed in the following way:

  1. In the first place, since the exponent is negative, and the rational base, the latter will be taken as the denominator of the unit, which will then make the exponent change its sign to positive.
  2. Next, in order to stop seeing the fraction as a denominator, its terms should be reversed.
  3. The inverted power will be elevated to the indicated exponent, of positive sign.
  4. If possible, simplify the fraction obtained.

Also, the way to solve this operation could be expressed mathematically in the following way:

Examples of how to solve powers of rational base and negative exponent

However, always the most efficient way to study the correct way to solve an operation will be through the revision of some examples that will allow us to observe in practice how the theory is fulfilled. The following are some cases of rational base and negative exponent powers:

Example 1

Resolve the next operation:

Since the fraction is elevated to a negative exponent, the method indicated in Mathematics should be applied for this type of cases, seeking then to invert the elements of the operation:

At this point, we will seek to solve the rational power operation that has been obtained:

Since there is no number that can divide both elements, it is considered that the fraction isn´t simplifiable, and therefore it will be taken as the solution of the operation.

Example 2

Resolve the next operation:

However, it will not always be necessary to express each of the steps leading to the inversion of the elements of the fraction once it serves as the base for a negative number. In this way it will be possible, every time there is an operation of this type, to simply invert the exponents, and place the positive exponent:

Once this has been done, the rational base power solution must be continued:

Once the answer is obtained, we will try to simplify it:

In this case, since the simplified fraction has the unit as denominator, it will then be taken as an integer, as well as the solution of the operation:

Example 3

Resolve the next operation:

In spite of the fact that the exponent is one, and this already implies that the result will be the same fraction that serves as base, the fact that it has a negative sign, will imply that the fraction must be inverted, in order to make the unit acquire the positive sign:

In doing so, we obtain a power of rational base and exponent 1, which will then be resolved as the mathematical law indicates in this regard:

Since the fraction cannot be simplified, this fraction will be taken as the final solution of the rational and exponent negative base power operation.

Example 4

Resolve the next operation:

In this case we will have a rational base power, elevated to a second exponent. Consequently, the operation will begin by multiplying both exponents, in order to convert them into a single value, to which the fraction that serves as the base for the first exponent will also be elevated:

In doing so, a rational base and negative exponent power operation is obtained, so it will be necessary to invert the elements -numerator and denominator- of the fraction, so that the exponent loses its negative sign:

Once this step has been completed, each element of the fraction should be elevated to the positive exponent:

It is then obtained a fraction with elements of elevated values, reason why the advisable thing will be to try to obtain the simplest form, dividing each one of these values in a number that can divide them to both:

Considering that the fraction obtained has a denominator equal to the unit, it can be expressed as an integer, which is also considered as the solution to the operation:

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Examples of how to solve a power of rational base and negative exponent
Source: Education  
October 25, 2019


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