Examples of how to rationalize denominators where there are additions and subtractions

Perhaps the most advisable thing to do, before explaining the correct way to rationalize a denominator, in which the existence of additions and subtractions can be appreciated, is to take a moment to briefly review the very definition of this operation.

Rationalization of denominators

However, it is also necessary to first bring into chapter the definition of fraction, which is understood by Mathematics as an expression, by means of which it takes account of fractional numbers, or what is equal to non-exact or non-integer quantities. Likewise, Mathematics points out that fractions will be composed of two elements: numerator, whose function is to occupy the upper part of the expression, indicating how many parts of the whole it represents; and denominator, which constitutes the lower part of the fraction, being used to show into how many parts the whole is divided.

As for the Rationalization of denominators, the different sources indicate that this is a mathematical operation, which takes place when a fraction has at least one radical expression in its denominator, and which consists precisely in getting this number out of the radical, in order to be able to continue simplifying the fraction.

Steps to rationalize denominators where there are additions and subtractions

Likewise, the mathematical discipline indicates that when choosing the correct form in which an operation of this type must be solved, first of all it must be verified if the denominator is composed by an expression that has neither additions nor subtractions, or on the contrary it does, because this will determine which is the procedure to follow. If the denominator includes this type of operations, the following steps must be followed:

  1. It will be determined which is the conjugated expression of the denominator, which consists of an expression that has the same elements, but the opposite sign.
  2. Multiply the number that serves as numerator by the conjugated expression of the denominator.
  3. Likewise, multiply the number that acts as denominator by its conjugated expression, which should lead to take out the radical elements that act as denominator from the radical sign.

This mathematical operation can be expressed in the following way:

Examples of how to rationalize denominators where there are additions and subtractions

However, the most efficient way to study how to solve this type of operations may be through examples that allow us to see how each of these steps are fulfilled, as shown below:


Rationalise the next denominator:

At the moment of beginning to solve the proposed operation, the denominator must then be revised, which makes it be observed that there is the presence of a sum in it. Therefore, at the moment of rationalizing it, both elements of the fraction must be multiplied by the conjugated expression of the denominator, understood as an expression composed by literal equals but inverse sign, and that will then lead to leave the radical sign:

The denominator of this fraction is then considered rationalized, and the result can be expressed in the following way:

Example 2

Rationalise, in the following fraction, the denominator:

In this case, the multiplication of each element by its conjugated expression will also be applied (that is, by expressions that even when they present the same literals, have the inverse sign of the original) in order to achieve that the numbers found in the denominated and wrapped by the radical sign come out of them:

Once this result is obtained, the denominator of the fraction can be considered rationalized, so it will be valid to express the result in the following way:

Example 3

Rationalise the denominator of the next fraction:

There may also be cases where in addition to an addition or subtraction operation in the denominator, there is more than one element in the numerator. Before this, each element of the fraction will be multiplied equally by the conjugated expression of the denominator:

At this point, the pertinent additions and subtractions between similar radical elements will be resolved, that is, those that share the same radicands and indexes, in order to reduce from four elements in the numerator to two of them. On the other hand, in the denominator –whose elements have already been taken out of the radicals, it will be possible to continue with the proposed subtraction:

Once the pertinent additions and subtractions have been solved, the denominator is equal to the unit, so it is not necessary to express this quantity in the form of a fraction, so it can be annotated in the following way:

Consequently, the denominator of this fraction can be considered to have been rationalized:

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phoneia.com (October 24, 2019). Examples of how to rationalize denominators where there are additions and subtractions. Bogotá: E-Cultura Group. Recovered from https://phoneia.com/en/education/examples-of-how-to-rationalize-denominators-where-there-are-additions-and-subtractions/