Learn Simplifying Radical Expressions

In this video, we are going to simplify radical expressions. Note that a perfect square number has a rational square root. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

$\sqrt{4}=2$ because $2^2=4$ ,
$\sqrt{9}=3$ because $3^2=9$ ,
$\sqrt{16}=4$ because $4^2=16$ , and so on…

There are other radical expressions other than perfect square ones, to which they may not have a rational square root.
$\sqrt{5}$ and $\sqrt{20}$ , and $\sqrt{32}$ are some examples.

However, having an irrational square doesn’t mean that the radical expression cannot be simplified.
$\sqrt{5}$ cannot be simplified because it is in its simplest state.

For $\sqrt{20}$ , 4 and 5 are factors of 20. Choosing factors that are perfect squares make the simplifying process easier.
$\sqrt{20}$ = $\sqrt{4}\times\sqrt{5}$
$\sqrt{4}$ is 2, so $\sqrt{4}\times\sqrt{5}$ is really just $2\sqrt{5}$

For $\sqrt{32}$ , 4 and 8 are factors of 32.
$\sqrt{32}$ = $\sqrt{4}\times\sqrt{8}$
$\sqrt{4}$ is 2, so $\sqrt{4}\times\sqrt{8}$ is $2\sqrt{8}$
$\sqrt{8}$ can be further simplified into $\sqrt{4}$ and $\sqrt{2}$ , or $2\times\sqrt{2}$
In the end, $\sqrt{32}$ is $4\sqrt{2}$

Another way of simplifying $\sqrt{32}$ is to use $\sqrt{16}$ and $\sqrt{2}$ as the factors.
Since 16 is a perfect square, $\sqrt{16}\times\sqrt{2}$ can be written as $4\sqrt{2}$ , leading to the answer immediately.

Now let’s try three more examples:
$\sqrt{75}$
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 $5\sqrt{3}$

$\sqrt{48}$
4 and 12 go into 48
$\sqrt{4}$ can be broken down into 2 and $\sqrt{12}$ can be broken down into $\sqrt{4}$ and $\sqrt{3}$
The square root of 4 is 2, $2\times2\times\sqrt{3}$ , so we have $4\sqrt{3}$

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 $4\sqrt{3}$ , same answer

$\sqrt{63}$
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 $3\sqrt{7}$

Examples of Simplifying Radical Expressions

Example 1

$\sqrt{72}$ ,

36 and 2 are factors of 72.
Now, we have:
$\sqrt{72}$ = $\sqrt{36}\times\sqrt{2}$
$\sqrt{36}$ is $6$
Therefore, the answer is
$6\sqrt{2}$

Example 2

$\sqrt{98}$ ,

49 and 2 are factors of 49.
Now, we have:
$\sqrt{98}$ = $\sqrt{49}\times\sqrt{2}$
$\sqrt{49}$ is $7$
Therefore, the answer is
$7\sqrt{2}$

Video-Lesson Transcript

Let’s go over how to simplify radical expressions.

Let’s start with some basic ones.

$\sqrt{4} = 2$ because $2^2 = 4$ ; exponent is the inverse of square root

$\sqrt{9} = 3$

$\sqrt{16} = 4$

$\sqrt{25} = 5$

$\sqrt{36} = 6$

$\sqrt{49} = 7$

$\sqrt{64} = 8$

$\sqrt{81} = 9$

$\sqrt{100} = 10$

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:

$\sqrt{5}$

There is no integer that you can multiply by itself that will give you $5$ .

Other examples of irrational are

$\sqrt{20}$

$\sqrt{32}$

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

So let’s take a look at that next.

$\sqrt{5}$ cannot be reduced

Next, $\sqrt{20}$ .

What numbers go into $20$ ?

Well, $4$ and $5$ .

We want to pick these numbers since $4$ is a perfect square.

So it will be

$\sqrt{20}$

$\sqrt{4} \sqrt{5}$

$2 \sqrt{5}$

Now, let’s see

$\sqrt{32}$

$\sqrt{4} \sqrt{8}$

$2 \sqrt{8}$

$2 \sqrt{4} \sqrt{2}$

$2 \times 2 \sqrt{2}$

$4 \sqrt{2}$

Now, let’s think about this.

We were able to break $\sqrt{32}$ twice.

First into $4$ and $8$ . Then $8$ into $4$ and $2$ .

Probably there’s a bigger number that we could’ve picked that goes into $\sqrt{32}$ .

That is $16$ .

So let’s do it again.

$\sqrt{32}$

$\sqrt{16} \sqrt{2}$

$4 \sqrt{2}$

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.

$\sqrt{75}$

What perfect square goes into $75$ ?

$25$ does.

$\sqrt{75}$

$\sqrt{25} \sqrt{3}$

$5 \sqrt{3}$

Now, $\sqrt{48}$ .

What’s the largest perfect square that goes into $48$ ?

Let’s try this

$\sqrt{48}$

$\sqrt{4} \sqrt{12}$

$2 \sqrt{12}$

$2 \sqrt{4} \sqrt{3}$

$2 \times 2 \sqrt{3}$

$4 \sqrt{3}$

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.

$\sqrt{48}$

$\sqrt{16} \sqrt{3}$

$4 \sqrt{3}$

We still got the same answer as above.

Now, let’s look at $\sqrt{63}$

What goes into $63$ and is a perfect square?

$\sqrt{63}$

$\sqrt{9} \sqrt{7}$

So, our answer is

$3 \sqrt{7}$