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
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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:
2- When in the denominator there is an addition or subtraction in which a term or both is a SQUARE root, such as:
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:
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:
To start eliminating the denominator root, we multiply it by another one that has the same index:
The radicando of the new root must be a power with the same base as the current root:
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:
And we put it in the new root:
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:
And from here, we begin to perform the multiplication in the denominator. First, we join the two roots into one:
We multiply the powers, keeping the base and adding the exponents:
And finally, the root is annulled with the exponent of the radicando, which was what we were looking for from the beginning:
Let’s see another example of rationalizing radicals, to make it even clearer:
First, we multiply by a root with the same index:
Within the new root, the potency must have the same base:
And its exponent results from subtracting the index minus the exponent of the radicand from the current root:
We also multiply the numerator by the root obtained:
And we begin to multiply the roots in the denominator, unifying them into a single root:
Now we multiply the powers we have inside the new root, keeping the base and adding the exponents:
And finally the denominator root is annulled:
I will now explain the second case of rationalizing radicals.
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:
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:
It would be respectively:
As you can see, we have only changed the central sign.
Knowing this, let’s explain a step-by-step example:
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:
Now we apply the property of the difference of squares in the denominator:
Y solve the squares:
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:
Let’s go with another example:
We begin by multiplying the numerator and denominator by the conjugate of the denominator:
We apply the remarkable product of the square difference in the denominator;
We solve the squares and cancel the term that had the root:
And now we operate to simplify the expression: