Examples of how to rationalize denominators

It is probable that the most pertinent thing, previous to approach each one of the exercises that can serve as example, to the correct form in which any rationalization of denominators must be resolved, is to take into account the own definition of this operation, to be able to understand each one of the cases from its precise context then.

Fundamental definitions

However, it will also be necessary to remember that for Mathematics a fraction is a mathematical expression, used to account for rational numbers, that is, for non-exact or non-integer quantities. Likewise, this discipline assumes the fraction as an expression composed of two elements: numerator, which will be the element found in the upper part in order to indicate how many parts of the whole the fraction represents; and denominator, an element that will occupy the lower part and will indicate how many parts that whole is divided into.

In this order of ideas, we can then say that the Rationalization of denominators is the mathematical operation applied to every fraction, where there is presence of a denominator where there are radical numbers, in order to be able to take them out of the radical and then obtain a fraction where the denominator has no presence of denominator, and that can then continue to be simplified.

Steps to rationalize a denominator

Likewise, Mathematics has indicated which are the steps to follow when carrying out the rationalization of a denominator. However, this discipline has distinguished between two possible cases, a difference that will basically depend on whether in the denominator there is, in addition to a radical number, the absence or presence of additions and subtractions, which will indicate different processes:

  • If there is no addition or subtraction in the denominator: if there is only one rational number in the denominator, the mathematical discipline indicates that to rationalize it it will be necessary to simply multiply each element of the fraction by the radical number that serves as denominator, thus extracting the element from the root, which in turn can be expressed in the following way:

  • If there are additions and subtractions in the denominator: on the contrary, if there are other operations in the denominator besides the radicals, specifically that of addition and subtraction, then when rationalizing this element it will be necessary to multiply both denominator and denominator by the conjugated expression of the denominator, which will simply be the same operation, with the same elements but different signs. On the other hand, this operation can be expressed in the following way:

Examples of rationalization of denominators

However, the most appropriate way to study this mathematical operation may be through the exposition of some examples, where you can see in a practical way how each case is solved:

Example 1

Rationalise the next denominator:

The first thing to do is to review the denominator, in order to determine whether or not there are additions or subtractions in it, since this will determine how the rationalization operation should be performed. In this case, the denominator is composed only of one radical, so it will be enough to multiply each element of this operation by the radical that constitutes the denominator:

Example 2

Rationalise the next denominator:

Once the denominator of this fraction has been revised, it will then be found to be made up of two radicals that are summing up. Consequently, having a sum in the denominator, it is assumed that this rationalization must be carried out by multiplying each element of the fraction by the conjugated expression of the denominator, that is, with an expression conformed by the same elements, but related to the opposite sign:

Example 3

Rationalise the next denominator:

Not every time you face a fraction that has to rationalize its denominator will have a numerator composed of a single element. However, that in this element -i.e. in the numerator- there are additions or subtractions, in reality it does not make any difference, so equally, at the time of giving solution to the approach given by the exercise, and seeing that in the denominator there are no additions or subtractions, we will proceed to multiply each element by the radical that serves as denominator:

Example 4

Rationalise the denominator that makes up the fraction offered below:

At the moment of beginning to solve this operation, the denominator will be revised, finding that in it exists a subtraction of radicals. Therefore, it is understood that at the moment of beginning to solve this rationalization operation, it will be necessary to multiply each element of the fraction by the conjugated expression of the denominator, that is to say, by the same elements but with the operation with the opposite sign:

Example 5

Rationalise the denominator of the next fraction:

Likewise, not all fractions where the denominator must be rationalized will have a numerator composed of a single element. However, this does not influence the rationalization, which considering that it is a denominator where there is a subtraction must be multiplied by its conjugated expression:

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