Examples of how to solve powers with rational base

Perhaps the most convenient thing, before approaching each one of the exercises that can serve as an example to the correct way in which the exercises of Powers of rational base must be solved, is to revise briefly the own definition of this operation, in order to understand each one of the cases presented within its just mathematical context.

Rational base powers

However, it may also be prudent to recall briefly that fractions are a type of mathematical expression, used to account for rational numbers, that is, non-integer or non-exact quantities, which are equally made up of two elements: the numerator, which will occupy the top of the fraction, showing how many parts of the whole represent the fraction; and the denominator, which occupies the bottom, will indicate into how many parts the whole is divided.

With regard to the operation of Rational-based Powers, it should be remembered that Potentiation is an abbreviated multiplication procedure in which a number -which serves as base- multiplies itself, as many times as a second number points out -which will act as exponent- in order to obtain a result, called power. In the case of rational base Powers, it will be this same operation, but instead of having an integer as a base, they have a fraction. Therefore, the fraction offered should be multiplied by itself as many times as the natural number given as exponent indicates.

Steps to solve a rational base power

However, although rational base Power can then be defined as an abbreviated multiplication of the fraction, Mathematics also suggests that instead of multiplying this base by itself, as often as the exponent tells you, it would be best to elevate each element of the fraction to this one, then applying what is known as the formula for solving rational base powers. Consequently, when solving this type of operations, the following steps should be followed:

  1. Specify which is the base and which is the exponent of the operation.
  2. Apply the mathematical formula for these cases, and elevate each member of the fraction to the indicated exponent.
  3. Review the product obtained, and verify if it is possible to continue simplifying, in order to obtain the simplest form that the fraction can have.

In this way, the formula to be applied in this type of cases can be expressed mathematically in the following way:

Examples of how to solve rational base powers

However, the best way to approach the study of this operation may be through the exposition of some exercises, which allow us to see in practice how each of these steps are fulfilled. The following are some examples of how to resolve powers on a rational base:

Example 1

Resolve the next operation:

Once the operation has been proposed, simply apply the pertinent mathematical formula, elevating each of the elements of the fraction to the cube:

Once this result is obtained, it will be impossible to find a number that can simplify both elements of the fraction, so it is considered that the final result of the operation has been achieved.

Example 2

Find the power of the next operation:

In the same way, and as should be done in all cases of rational base powers, once the potentiation operation has been proposed, each element of the fraction must be elevated to the given exponent:

Before this result, and considering that both elements of the fraction are constituted by even numbers, a number will be searched then that can simplify this fraction until its simplest expression:

Once this result has been obtained, it is assumed that the simplest form of the expression has been found, and the rational base power operation can then be interpreted as resolved:

Example 3

Resolve the next rational base power operation:

It can also happen that the fraction that serves as base for the rational base power operation is elevated to an exponent equal to zero. In this type of cases -and in fact in all without exception, independently of the quantity expressed by each element of the fraction- the mathematical property must be applied that dictates that every number elevated to an exponent equal to 0, gives as a result the unit. Therefore, it is necessary to apply the mathematical property that dictates that every number elevated to an exponent equal to 0 results in the unit:

Example 4

Resolve the next operation:

Likewise, if the proposed operation includes a fraction that has to be elevated to an exponent equal to the unit, then the mathematical property that exists in this respect must be brought to chapter, and that indicates that every fraction that elevates to the unit must give as a result itself. This law will always apply, regardless of whether it is an integer or a fractional number:

Example 5

Resolve the next operation:

Within the diversity of approach that may exist in reference to the rational base powers, there will also be those that pose negative rational bases. Faced with this, it is necessary to remember the mathematical Law that is applied in this type of cases:

  1. If the base is positive, its power is also positive.
  2. If the base is negative and its exponent pair, the power will be positive.
  3. If the base is negative and the exponent is odd, the power will be negative.

In this example, we then have an odd base elevated to the cube, which will imply that each element of the fraction is elevated to this exponent, and that according to the mathematical Law the result is also assumed as negative:

Once this result is obtained, and if it is impossible to continue simplifying the fraction, then the rational base power will be considered resolved.

Example 6

Solve the next rational base power:

Finally, it will be important to point out that there can also exist powers of rational base elevated to negative exponents, in which case it will be necessary to apply the property that dictates that in case of being before an operation of this type, it will have to convert the fraction and its negative exponent in the denominator of the unit, in order that the exponent changes its sign to positive, likewise later the terms of the fraction will have to be inverted, elevating it to the positive exponent:

Once this result has been achieved, the operation of rational base powers will continue:

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

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