Do you want to learn how to multiply and divide radicals? I’ll explain it to you below with step-by-step exercises.
After seeing how to add and subtract radicals, it’s up to the multiplication and division of radicals.
Next I’ll also teach you how to multiply and divide radicals with different indexes.
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Product and radical quotient with the same index
In order to multiply radicals with the same index, the first property of the roots must be applied:
Let’s look at it better with an example:
We have a multiplication of two roots. First of all, we unite them in a single radical applying the first property:
We have already multiplied the two roots. From here we have to operate to simplify the result. To do this, we multiply the powers within the radical by adding the exponents:
And finally, we extract factors out of the root:
The quotient of radicals with the same index would be resolved in a similar way, applying the second property of the roots:
To make this radical quotient with the same index, we first apply the second property of the roots:
Once the property is applied, you see that it is possible to solve the fraction, which has a whole result.
To finish simplifying the result, we factor the radicand and then the root will be annulled with the exponent:
That said, let’s go on to see how to multiply and divide roots that have different indexes.
Multiplication and division of radicals of different index
To understand this section you have to have very clear the following premise:
So how do you multiply and divide the roots that have different indexes?
Well, you have to get them to have the same index.
Before telling you how to do it, you must remember the concept of equivalent radical that we saw in the previous lesson. Therefore, since we can modify the index and the exponent of the radicando without the result of the root varying, we are going to take advantage of this concept to find the index that best suits us.
To obtain that all the roots of a product have the same index it is necessary to reduce them to a common index, calculating the minimum common multiple of the indexes.
When modifying the index, the exponent of the radicand will also be affected, so that the resulting root is equivalent to the original one.
With the new common index, indirectly we have already multiplied the index by a number, so we must know by which number the index has been multiplied to multiply the exponent of the radicand by the same number and thus have a root equivalent to the original one.
We calculate this number with the following formula:
Once calculated, we multiply the exponent of the radicando by this number.
It is exactly the same procedure as for adding and subtracting fractions with different denominator.
When we have all the roots with the same index, we can apply the properties of the roots and continue with the operation.
Let’s see it with several examples
Example of multiplication of radicals with different index
Let’s start with an example of multiplying roots with the different index
The first step is to calculate the minimum common multiple of the indices:
This will be the new common index, which we place already in the roots in the absence of the exponent of the radicando:
Now we must find the number by which the original index has been multiplied, so that the new index is 12 and we do it dividing this common index by the original index of each root:
That is to say, the index of the first root has been multiplied by 4, that of the second root by 3 and that of the third root by 6. Therefore, by those same numbers we are going to multiply each one of the exponents of the radicands:
We multiply exponents:
And we already have a multiplication of roots with the same index, whose roots are equivalent to the original ones.
We follow the procedure to multiply roots with the same index. You will see that it is very important to master both the properties of the roots and the properties of the powers.
We unite the three roots into one:
Inside the root there are three powers that have different bases. As they are, they cannot be multiplied, since only the powers with the same base can be multiplied.
In order to find the powers that have the same base, it is necessary to break them down into prime factors:
Once decomposed, we see that there is only one base left. Then, we eliminate parentheses and finally, we can add the exponents keeping the base:
We already have the multiplication. Now let’s simplify the result by extracting factors out of the root:
And finally, we simplify the root by dividing the index and the exponent of the radicand by 4 (the same as if it were a fraction).
Example of radical division of different index
Let’s see another example of how to solve a root quotient with a different index:
First, we reduce to a common index, calculating the minimum common multiple of the indices:
We place the new index in the roots and prepare to calculate the new exponent of each radicando:
We calculate the number by which the original index has been multiplied, so that the new index is 6, dividing this common index by the original index of each root:
We multiply the exponents of the radicands by the same numbers:
We already have the equivalent roots with the same index, so we start their division, joining them in a single root:
We now divide the powers by subtracting the exponents:
And to finish, although if you leave it that way nothing would happen, we can leave the exponent as positive, passing it to the denominator:
Example of product and quotient of roots with different index
Let’s solve a last example where we have in the same operation multiplications and divisions of roots with different index.
In addition, we will put into practice the properties of both the roots and the powers, which will serve as a review of previous lessons.
We have some roots within others. Therefore, the first step is to join those roots, multiplying the indexes. First we put the root fraction as a fraction of roots:
And now we can multiply their indices:
We are left with an operation with multiplication and division of roots of different index.
We reduce them to a common index, calculating the minimum common multiple:
We place the new index and also multiply the exponents of each radicando:
We multiply the numerators and denominators separately:
And finally, we proceed to division, uniting the roots into one. Within the root there remains a division of powers in which we have two bases, which we subtract from their exponents separately.
We have left the powers in the denominator so that they appear with a positive exponent.