Examples of how to rationalize denominators in which there are no additions or subtractions

Before approaching each one of the exercises that can serve as an example to the correct way in which a denominator must be rationalized, in which there are no additions or subtractions, perhaps the best thing to do is to take a moment to revise the definition of this operation, in order to be able to understand each one of the cases from its precise mathematical context.

Rationalization of denominators (without additions or subtractions)

However, it may also be pertinent to remember first that Mathematics defines the fraction as a mathematical expression, used to represent fractional or rational numbers, i.e., numerical elements that in turn symbolize non-exact or integer quantities. Likewise, this discipline warns that fractions will always and without exception be composed of two elements: the numerator, which will occupy the upper part, indicating how many parts of the whole the fraction represents; and the denominator, which will be located below, and will indicate how many parts the whole is divided into.

For its part, the Rationalization of denominators will be the operation that will be carried out whenever we want to continue simplifying a fraction, where there is the presence of a radical denominator. In this case, Mathematics indicates thatthis operation must be carried out in order to remove from the radicals the numbers that constitute the denominator. However, the correct way to solve this type of operation will consist in whether or not in this element of the fraction, in addition to the radicals, there are additions or subtractions.

Steps to rationalize a denominator where there are no additions or subtractions

In the case where denominators are presented, compounds a number radica, which does not establish any type of addition or subtraction with any other element, radical or not, Mathematics indicates that for its rationalization the following steps must be followed:

1. The numerator should be multipliedby the radical that acts as denominator.
2. The denominator will be multiplied by itself, obtaining that it rises to the square, managing then to leave the radical.
3. The fraction obtained after rationalization is simplified.

This mathematical operation can be expressed in the following way:

Examples of rationalization of denominators where there are no additions or subtractions

However, perhaps the most appropriate way to approach a study of this mathematical operation is through some examples that allow us to see in practice how each of the steps inherent to its solution are fulfilled. Here are some of them:

Example 1

Rationalise the next denominator:

When wanting to rationalize a denominator where a radical is present, the element should be reviewed to verify that there are no additions or subtractions. Once this has been done, it may be decided that, in accordance with the nature of the denominator, the correct way to rationalizeit will be by multiplying each element of the fraction by the radical that constitutes the denominator of the fraction:

Once the operation has been presented in this way, the square of the denominator is finally resolved, which will make the rationalization concrete, succeeding in removing the element from the radical sign:

Exercise 2

Rationalise the next denominator:

Once it has been determined that the denominator numbers are radical, but have no addition or subtraction in them, then the operation can be solved. However, we must also see the numerator, where there is an addition, which must be solved before rationalization, but which in reality does not represent any change in the way of solving this type of operations:

Exercise 3

Rationalise the next denominator:

Likewise, it may happen that in the denominator to be rationalized there is no presence of addition or subtraction, but in the numerator there is some of these operations, as well as radicals. However, the fact that this exists in the numerator does not change the way of solving these operations, which will be solved by multiplying each element by the radical that constitutes the denominator:

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