In this video, we are going to simplify radical expressions. Note that a perfect square number has a rational square root.

because ,

because ,

because , and so on…

There are other radical expressions other than perfect square ones, to which they may not have a rational square root.

and , and are some examples.

However, having an irrational square doesn’t mean that the radical expression cannot be simplified.

cannot be simplified because it is in its simplest state.

For , *4* and *5* are factors of *20*. Choosing factors that are perfect squares make the simplifying process easier.

=

is 2, so is really just

For , *4* and *8* are factors of *32*.

=

is 2, so is

can be further simplified into and , or

In the end, is

Another way of simplifying is to use and as the factors.

Since 16 is a perfect square, can be written as , leading to the answer immediately.

Now let’s try three more examples:

*25* is a perfect square that goes into *75* and the other factor is *3*

The square root of *25* is *5* so the final answer is

*4* and *12* go into 48

can be broken down into *2* and can be broken down into and

The square root of *4* is *2**, *, so we have

Now let’s do the same problem by using the perfect square

*16* is a perfect of *48* and the accompanying factor is *3
*The square root of

*16*is

*4*, so we have , same answer

*9* is a perfect square that goes into *63* and the other factor is *7*

The square root of *9* is *3* so the final answer is

## Video-Lesson Transcript

Let’s go over how to simplify radical expressions.

Let’s start with some basic ones.

because ; exponent is the inverse of square root

These are all known as perfect squares.

The rational answer exists.

These are rational expressions. But sometimes we have the square root of a number that we can’t do the square root of. That is called an irrational.

For example:

There is no integer that you can multiply by itself that will give you .

Other examples of irrational are

But even if they are irrational, doesn’t mean that we cannot reduce them.

So let’s take a look at that next.

cannot be reduced

Next, .

What numbers go into ?

Well, and .

We want to pick these numbers since is a perfect square.

So it will be

Now, let’s see

Now, let’s think about this.

We were able to break twice.

First into and . Then into and .

Probably there’s a bigger number that we could’ve picked that goes into .

That is .

So let’s do it again.

And we came up with the same answer.

Notice that if we start off with the biggest possible perfect square that goes into a number, we’ll get the answer immediately.

If we don’t, like what we did prior to this, we’ll get the same answer but took us some more steps.

Let’s have three more examples.

What perfect square goes into ?

does.

Now, .

What’s the largest perfect square that goes into ?

Let’s try this

As you can see, we break this down twice.

So we probably didn’t get the biggest perfect square in the beginning.

Let’s do it again.

We still got the same answer as above.

Now, let’s look at

What goes into and is a perfect square?

So, our answer is