Rationalization of denominators

It is likely that the best way to address an explanation of the correct way to resolve a denominator Rationalization operation is to pre-review some definitions, which will allow you to understand this procedure in your precise context.

Fundamental definitions

In this regard, it may also be relevant to delimit this revision to two specific notions: Fractions and Radication, as these are expressions and operations, directly related to the procedure by which the Rationalization of Denominators. Here are each of these concepts:


In this way, it will begin by saying that Mathematics has explained fractions as one of the two possible types of expression with which fractional or rational numbers count, that is, that fractions will be mathematical expressions that serve to represent non-exact or non-whole amounts. Likewise, fractions are pointed out by most sources as an expression composed of two elements, each of which has been explained as follows:

  • Numerator: First, the Numerator will be assumed as the numeric element that occupies the top of the fraction. Its mission is basically to indicate how many parts of the whole have been taken, or are represented by the fraction.
  • Denominator: On the other hand, the Denominator can be understood as the element located at the bottom of this expression. It has the task of indicating in how many parts the whole is divided, of which the numerator points out only a few, and sometimes all.


In another order of ideas, it will also be necessary to dwell on the concept of Radicación, a mathematical operation whose main purpose, as the different authors point out, will be to determine what is the number, which once raised to the index that the original proposed resulting in the establishment that it also raises from the outset. Consequently, the Radication could also be assumed as a reverse expression of the Empoweror, since if the operation were to be raised in these terms, then the basis would be sought.

Rationalization of denominators

Once each of these concepts has been revised, it will be easier to delve into the definition of Rationalization of denominators, mathematical operation, whose main motivation is to remove from the root the number that serves as a denominator in a fraction, to in order to reduce the mathematical expression to a smaller form, which could not be done by calling a radical.

However, the operation involving the rationalization of denominators may encounter two realities, depending on whether the denominator does not have sums or subtractions in its configuration, or if it does present them, cases involving ways in which different from taking on the process of rationalizing denominators. Here are each of the possible options:

If no sums or subtractions are presented in the denominator

If the denominator of a fraction is a radical, which does not add or subtract with any other element, and you wish to apply a rationalization procedure, each of the elements of the fraction must be taken and multiplied by the radical that constitutes that of nominator, in order to get a square of this, to allow him to get out of the root. This procedure could be expressed mathematically as follows:

However, it may be necessary to provide a concrete example of how to proceed with a Rationalization of denominators, involving an element in which neither additions nor subtractions, such as the following, are presented:

If the denominator does have additions or subtractions

On the contrary, it may also be the case that the Denominator that you want to rationalize counts itself with addition or subtraction operations, so in this case it will be necessary to multiply each of the elements of the fraction, that is, the numerator and denominator by the conjugated expression of the denominator, which will correspond to the following logics:

If the expression is (a – b) the conjugate expression will be (a + b)

If the expression is (a + b) the conjugate expression will be (a – b)

Consequently, the procedure for rationalising denominators, if it has an element where there is addition or subtraction, may be expressed mathematically as follows:

  • If, on the other hand, the denominator considers a subtraction, it would proceed as follows:
  • It is assumed, moreover, that the logic of multiplying a sum or subtraction by its conjugated expression will imply the following:

a + b . (a – b)=  a2 – b2

However, it may also be necessary to set out a specific example, which allows you to see in a practical way how each of the steps involved in rationalizing a denominator involving additions or subtractions is present, as can be seen below:

Rationalize the denominator in the following fraction:

To solve this operation, each element must then be multiplied by the conjugate expression of the denominator:

This will be taken as the final result of the operation of rationalization of denominators, since the mission of extracting the elements of the denominator of the radicals that surrounded them has been fulfilled.

Picture: pixabay.com

Rationalization of denominators
Source: Education  
September 30, 2019

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