Examples of how to express rational radicals in forms of powers

Perhaps the best thing before approaching each of the cases that can serve as an example of the operation consisting of expressing a rational radical in the form of power, is to revise the very definition of this operation, in order to try to understand this procedure within its precise mathematical context.


Expression of a rational radical in the form of power

However, before continuing with the definition of this operation, we must also remember that a rational radical will be any mathematical expression in which there is a rational number wrapped by a radical sign, that is, any radical expression that has a fraction as its root. On the other hand, power will also be a mathematical operation in which a number -which serves as base- multiplies itself as many times as a second element, called exponent, indicates. In the case of radical base powers, operations of this type will be based on a fraction.

For its part, Mathematics has indicated that the operation by means of which a rational radical is expressed in the form of power will be the procedure consisting in achieving that a radical, which has a fraction as its root, becomes a power, where both base and exponents are composed of rational elements. In this way, the radical sign is dispensed with, and the fraction goes from being a radicando to being a base.

Steps to convert a rational radical into a power

  1. Likewise, the mathematical discipline indicates that the appropriate way to perform this operation is composed of a series of steps, then described as follows:
  2. The index of the radical will be specified.
  3. The exponent to which the rational radical is elevated will be specified.
  4. The exponent of the rational radical will be taken as the numerator of the new exponent, while the index of the radical will become the denominator of this rational exponent.
  5. Leave the fraction that served as radicando in the form of base and raise to the rational exponent that has been formed with the exponent and index that originally affected the rational radical.

This operation can be expressed mathematically in the following way:

Examples of how to express a rational radical in the form of power

However, the best way to study this type of mathematical procedure may be through a series of examples, which will serve to see in a practical way how this operation is carried out correctly. Here are some of them:

Example 1

To express in the form of power the following rational radical:

In this case, you will have a rational radical of index 3, i.e. a cubic root, which in turn is squared. To convert this expression into a power, the exponent and the index must be taken, so that they act respectively as numerator and denominator of the rational exponent to which the fraction will be elevated:

Example 2

Express the following rational radical in the form of power:

For its part, in this example we will have a fourth root, and a rational radical that apparently is not elevated to any exponent. In this case, it is assumed that the exponent of the rational radical is implicit, and is constituted by the unit, which will then serve as numerator to the rational exponent that will have this rational radical when expressed in the form of a fraction, while the denominator of the exponent will be constituted by the 4 of the index:

Example 3

Express in the form of power the following expression:

When reviewing this rational radical, two circumstances will appear to be evaluated. First, the rational radical will not be elevated to an explicit exponent, so it will be considered equal to unity. On the other hand, the index of the radical is not indicated in written form either, so it will be considered equal to two, that is to say that it is a square root. Determined each one of these elements, then we will know how the rational exponent to which the fraction that serves as radicand must be conformed, to be expressed then as a power:

Example 4

To express in the form of power the following rational radical:

As in all cases involving this type of procedure, both the exponent and the index that are affecting the rational number that serves as the radicando must be specified. Consequently, it has to be a rational radical of index three (cubic root) which is also elevated to the cube. Consequently, when this step is finished, it is known then how the exponent to which the radicando will rise when it begins to work as power will have to be conformed:

In this case, a particularity is presented, because being conformed exponent and index by the same number, then a rational number is created as exponent that will be solved as one. When doing, the fraction is elevated to an exponent equal to 1, resolving the potentiation operation as indicated by the norm: every number elevated to an exponent equal to 1 is equal to itself:

The task of expressing a rational radical as a power is then considered fulfilled.

Image: pixabay.com

Examples of how to express rational radicals in forms of powers
Source: Education  
October 24, 2019


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