Root properties: Application and examples solved step by step

Knowing how to use root properties is essential to simplify operations with radicals correctly, as well as to perform operations with radicals.

So, next, we’ll look at the properties of the roots one by one, and how you have to apply them in your operations.

Property 1. Multiplication of roots with the same index

Multiplying two roots with the same index is the same as multiplying into a single root with that index:

For example, if you have a multiplication within a root, you can separate each factor and solve each root separately, to get the final result:

This property is going to be very useful for example to understand how to introduce or extract factors in a root.

Property 2. Division of roots with the same index

The same happens with the division of two roots with the same index. That division is equivalent to the root of the division.

The root of a division is equal to the division of the roots:

For example:

Very useful property also to simplify operations with roots.

Property 3. Root elevated to an exponent

Another of the properties of roots is when you have a root elevated to an exponent, it is equivalent to that exponent being inside the root elevating the radicand:

For example:

This property should not be confused, multiplying the exponent by the index. Be very careful. For that we already have the following property.

Property 4. Root of another root

One root of another root is equal to another root whose index is the product the two indexes:

For example:

This property can be applied with all levels of root roots. For example:

Property 5. Cancellation of the root

As you already know, the root is the opposite operation to the power.

So if you have a number or a variable elevated to an exponent that is within a root with the same index, the power with the root is cancelled out:

This property seems obvious, but when it is part of a much more complex expression, it is sometimes forgotten.

It is very useful for simplifying expressions when working with variables:

The index and exponent are cancelled and only the x is left

When you operate with numbers, you apply this property indirectly by obtaining the result of the root. For example, in this root:

The result is 2 because 2 elevated to the cube are 8.

If we apply this property and instead of writing 8, we put it as 2 elevated to the cube, we see that the 3 of the exponent index and of the index are annulled and only the 2 remains, which is the result of the root.

The properties of roots make more sense when they are used for some purpose, either to simplify an expression or to perform operations with roots.

We are going to solve a few exercises to familiarize you with them.

Examples of application of the properties of the roots

I’m going to solve with you some exercises so that you can see how the properties of the roots have to be applied:

Example 1:

When you have a root with more than one factor, as in this case, the first thing you have to do is to apply the property of the multiplication of roots and separate each factor in a root:

By having it in two roots, we can already obtain the result of the first root which is 2:

We are going to leave the second root like this, but we could extract factors.

Example 2:

Let’s see this other example with the root of a division:

To be able to solve this root, by means of the property of the division of roots, we convert it into the division of two roots to be able to solve each one of them separately:

Normally, when you see a fraction inside a root, the first step will always be to convert it to a division of two roots as we just did, but sometimes, you can simplify the fraction inside the root and it is not necessary to apply the property.

Example 3:

In this example we are going to have to apply more than one property:

The first step is to apply the property of the division, so we turn it into a division of two roots of index 3:

Now, if you notice, both in the numerator and in the denominator we have a root of more than one factor. Therefore the next step is to apply in each root the property of the multiplication and to separate it in two roots:

Once we can no longer separate into more roots, we can proceed to solve the one that has solution, as we just did.

It could be simplified even more, extracting factors.

Example 4:

In this other example of application of the properties of the roots, it is also necessary to apply more than one property, but the way to apply them works in an inverse way to the previous example:

In this case we have already separated roots. The root of 6 has no whole solution. We could apply the multiplication property of the roots of 8x and 12x, but we wouldn’t have any whole solution either.

What do you have to do then?

Let’s put all the roots together into one to see if we can simplify what’s left of this one root. We begin with the numerator of the fraction, joining the two roots into one and doing the multiplication that is left inside:

We now have the division of two roots. We unite it again in one, by means of the property of the division.

Inside the root we have a fraction that can be simplified and the result of that simplification has a whole solution:

At all times you must go looking for the whole solution of the root to apply the properties of the roots of multiplication and division in one direction or another.

You have to see if it is convenient or not to apply a property in one way or another, always looking for the whole solution of the root.