How to rationalize radicals. Exercises solved step by step

Do you know what rationalization is? RATIONALIZING RADICALS consists of eliminating the roots of the denominator.

To get rid of the roots of the denominator, it is obviously not enough to remove them from there, since we would modify the result of the expression. It is necessary to find an expression equivalent to the original but without roots in the denominator.

How do we do that? Well, let’s see it

How to rationalize radicals in expressions with radicals in the denominator

We have two cases in which we can rationalize radicals, i.e., eliminate the radicals from the denominator:

1- When in the denominator we have only one root (the index does not matter), as for example these expressions:

rationalize radicals

2- When in the denominator there is an addition or subtraction in which a term or both is a SQUARE root, such as:

rationalization of resolved radical exercises

We are going to see in detail, step by step, how to rationalize radicals in each of them.

How to rationalize radicals in expressions with a root in the denominator

To rationalize radicals when we only have one root in the denominator, we are going to eliminate it by applying the following property:

how to rationalize radicals

That is to say, we have to get the root to be annulled and for this we must make the exponent equal to the index.

To achieve this, we have to multiply the root of the denominator, by another root, such that when multiplying them, its result is another root that is annulled. And so as not to modify the result, the numerator is multiplied by the same root.

To do this, keep in mind that:

  • The roots must have the same index to be able to multiply them
  • When multiplying two powers with the same base, the exponents are added.
  • The exponent of the radicand of the new root must be complemented with the exponent of the radicand of the current root, so that its sum is equal to the index of the roots

Let’s take a slower look at it with an example:

rationalize examples

To start eliminating the denominator root, we multiply it by another one that has the same index:

rationalization exercises solved and explained

The radicando of the new root must be a power with the same base as the current root:

solved rationalization exercises

Y its exponent, must be such that when added with the exponent of the current root, its result is equal to the index. That is to say, the exponent of the radicand of the current root is 2 and we want another exponent (x) that when adding it, the result will be 5, therefore, that exponent will be 3:

steps to rationalize

And we put it in the new root:

rationalización de denominadores ejercicios resueltos

Now that we have obtained the root by which we have to multiply the denominator, so as not to modify the result of the expression, we must also multiply the numerator:

examples of rationalizing
And from here, we begin to perform the multiplication in the denominator. First, we join the two roots into one:

rationalize solved examples roots

We multiply the powers, keeping the base and adding the exponents:

rationalize exercises

And finally, the root is annulled with the exponent of the radicando, which was what we were looking for from the beginning:

rationalize radical exercises resolved

Let’s see another example of rationalizing radicals, to make it even clearer:

rationalize the denominator resolved exercises

First, we multiply by a root with the same index:

rationalización paso a paso

Within the new root, the potency must have the same base:

resolute rationalization exercises

And its exponent results from subtracting the index minus the exponent of the radicand from the current root:

rationalization of resolved denominators exercises

denominator rationalization exercises

We also multiply the numerator by the root obtained:

rationalizing exercises resolved

And we begin to multiply the roots in the denominator, unifying them into a single root:

rationalizes each expression

Now we multiply the powers we have inside the new root, keeping the base and adding the exponents:

examples of denominator rationalization

And finally the denominator root is annulled:

rationalize roots

I will now explain the second case of rationalizing radicals.

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Rationalizing radicals in expressions with an addition or subtraction of roots in the denominator

The second case of rationalizing radicals consists, as I indicated at the beginning of the lesson, in that in the denominator we have an addition or a subtraction of two terms, where at least one of them is a square root.

It is very important to note that it is a square root, since if it has another index, this procedure would not be valid and could not be rationalized.

As in the previous case, we have to eliminate the roots, but this time we will do it with the help of the property that a sum for a difference is equal to the difference of squares:

rationalize numerator and denominator at the same time

To do this, we have to multiply the numerator and denominator by the conjugate of the denominator.

To obtain the conjugate of an addition or a subtraction (of a binomial), we only have to change the central sign. For example, the conjugate of:

which is to rationalize in mathematics

It would be respectively:

conjugated radicals

As you can see, we have only changed the central sign.

Knowing this, let’s explain a step-by-step example:

which is to rationalize mathematics

To rationalize radicals in this expression, we multiply the numerator and denominator by the conjugate of the denominator, which we have obtained by changing the minus sign for a plus sign:

rationalizes the denominator of each expression

Now we apply the property of the difference of squares in the denominator:

ejercicios racionalalizar

How to rationalize a cubic root

Y solve the squares:

rationalizar raiz

As you have seen, when the roots were squared, they were annulled, which is what we were looking for, therefore, this procedure is only valid with square roots.

Now without roots, we continue to operate to simplify the expression:

rationalize denominators

Let’s go with another example:

rationalization exercises

We begin by multiplying the numerator and denominator by the conjugate of the denominator:

exercises to rationalize 4 eso

We apply the remarkable product of the square difference in the denominator;

rationalizacion de raíces ejercicios resueltos

We solve the squares and cancel the term that had the root:

conjugate of denominator

And now we operate to simplify the expression:

rationalizar ejercicios resuetos