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It is likely that the best way to approach an explanation of how a denominator Rationalization operation should be developed, when this element presents addition or subtractions, will be to make a preliminary review of some definitions, which will allow you to understand this procedure in its precise mathematical context.
Therefore, it may also be prudent to focus this conceptual review on three specific notions: Fractions, Rationalization of Denominators, and Conjugated Expression, because these are the expressions and operations, directly related to the operation aimed at addressing a denominator in which there is a presence of both radicals and additions or subtractions. Here are each of these definitions:
In this way, one can begin by saying that Mathematics has explained fractions as one of the two types of expression by which fractional numbers can be accounted for. Therefore, fractions will then be used to represent non-exact or non-whole quantities. Likewise, that discipline has pointed out that this mathematical expression consists of two elements, which have been explained in turn as follows:
- Numerator: First, you will find the Numerator, which has been described as the numerical element that occupies or makes up the top of the fraction, and whose mission is to identify how many parts of the whole have been taken, or are represented by the fraction.
- Denominator: On the other hand, the Denominator will be the element that constitutes the bottom of this mathematical expression. Your task will be to indicate how many parts the whole is divided, of which the numerator points out only a few.
Rationalization of denominators
Likewise, it will be necessary to pause for a moment on the concept of Rationalization of denominators, a mathematical operation that can be understood as the procedure by which any numerical element is taken from its radicals, which is tucked in with the radical sign, and which also constitutes a denominator.
Ergo, the Rationalation of denominators will extract from its radicals the elements that conform to the denominator of a fraction. This operation is performed for the purpose of driving the denominator to a state where it can be simplified.
Finally, it will also be prudent to cast lights on the definition of Conjugated Expression, a mathematical term that refers to the action of taking a mathematical expression or operation, preserving its elements or literals, but varying its sign on the contrary. This operation may be represented mathematically as follows:
a + b → a – b (conjugate expression)
a – b → a + b (conjugate expression)
Rationalization of denominators there are additions or subtractions
Once each of these concepts have been revised, it may then certainly be much easier to delve into the correct way in which one of the two cases that can arise when in the denominator that a denominator will be rationalized: where additions or subtractions exist between two or more numerical elements.
In this type of situation, the mathematical discipline states that the procedure to be followed will be to determine the conjugate expression of the denominator, and then multiply by this both the numerator and the denominator, an action that will allow to rationalize the latter element, that is, to extract from the radicals the numbers that constitute the denominator of the fraction, since the product of an expression by its conjugate will always be the square of the literals, which will allow the denominators to be taken out of the root. Situation that may be expressed as follows:
(a + b) . (a – b) = a2 – b2
For its part, the procedure indicated for resolving the Rationalization of denominators in the event that a sum or subtraction is present in this element shall be as follows:
If there is a sum in the denominator:
If there is a subtraction in the denominator:
Examples of how to rationalize denominators where additions and subtractions exist
However, perhaps the most efficient way to complete an explanation on the Rationalization of denominators, when these submit in their constitution addition or subtraction operations, will be through the presentation of a concrete example, which allows us to see in practice how each of the steps involved in the solution of this type of procedure are carried out, which will subsequently allow for the simplification of the expression. Then, a denoisade rationalization exercise:
Rationalize the denominator of the following fraction:
In order to solve this operation, the conjugate expression of the denominator will be searched, and each element of the fraction will be multiplied with it:
Once the denominator numbers of the radicals that arrote them have been extracted, Rationalization will be deemed to have been performed.
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September 30, 2019