In this video, we will multiply radical expressions. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

$\sqrt{3}\times\sqrt{3}=\sqrt{9}=3$
$\sqrt{5}\times\sqrt{5}=\sqrt{25}=3$

In other words,
$\sqrt{7}\times\sqrt{7}=7$
$\sqrt{12}\times\sqrt{12}=12$

Let’s try other examples:
$\sqrt{5}\times\sqrt{7}$ would equal to $\sqrt{35}$
$\sqrt{18}\times\sqrt{2}$ would equal to $\sqrt{36}$, which is a perfect square so it would be $6$

Let’s try the same problem by using another method:
$\sqrt{18}\times\sqrt{2}$ would equal to $\sqrt{36}$ can also be written as $\sqrt{9}\times\sqrt{2}\times\sqrt{2}$

For $\sqrt{15}\times\sqrt{45}$, it would equal to $\sqrt{675}$
25 goes into 675 twenty-seven times
Since 5 is the square root of 25, $\sqrt{675}$ can be written as $5\sqrt{27}$
This can be further simplified into $15\sqrt{3}$

## Examples of Multiplying Radical Expressions

### Example 1

$\sqrt{18}\times \sqrt{16}$
Use the rule to multiply the radicands $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$
$\sqrt{18\times 16}$
$\sqrt{288}$
Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors.
$\sqrt{144\times2}$
Simplify,
$12\sqrt{2}$

### Example 2

$\sqrt{4x}\times \sqrt{5x^3}$
Use the rule to multiply the radicands $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$
$\sqrt{4\times 5 \times x \times x^3}$
$\sqrt{4\times 5 \times x^4}$
Simplify,
$2x\sqrt{5}$

## Video-Lesson Transcript

Let’s go over how to multiply radical expressions.

$\sqrt{3} \times \sqrt{3} = \sqrt{9} = 3$
$\sqrt{5} \times \sqrt{5} = \sqrt{25} = 5$

So if you look, closely

$\sqrt{7} \times \sqrt{7} = 7$
$\sqrt{12} \times \sqrt{12} = 12$

Keep this in mind as we move forward in the lesson.

Let’s take a look at this

$\sqrt{5} \times \sqrt{7} = \sqrt{35}$

this cannot be simplified.

$\sqrt{18} \times \sqrt{2} = \sqrt{36} = 6$

But what if we reduce this first?

$\sqrt{9} \times \sqrt{2} \times \sqrt{2}$
$3 \times 2$
$6$

So you can multiply it across and get an answer.

Or reduce it first and multiply it out.

Let’s have this one

$\sqrt{15} \sqrt{45}$

I can multiply this out

$\sqrt{15} \sqrt{45} = \sqrt{675}$

Then we should break $675$ down.

But it’s a pretty large number.

$\sqrt{675}$
$\sqrt{25} \sqrt{27}$
$5 \sqrt{27}$
$5 \sqrt{9} \sqrt{3}$
$5 \times 3 \sqrt{3}$
$15 \sqrt{3}$

Let’s see what happens if we simplify the radical expressions first.

$\sqrt{15} \sqrt{45}$
$\sqrt{15} \sqrt{9} \sqrt{5}$
$\sqrt{15} \times (3) \sqrt{5}$
$3 \sqrt{75}$
$3 \sqrt{25} \sqrt{3}$
$3 \times 5 \sqrt{3}$
$15 \sqrt{3}$

We came up with the same answer.

The number of steps in the two methods is pretty much the same.

But I dealt with smaller numbers using the second method.

Let’s look back at a point here.

$\sqrt{15} \sqrt{45}$
$\sqrt{15} \sqrt{9} \sqrt{5}$

At this point, we know that $\sqrt{15}$ can’t be broken down.

But there’s a $\sqrt{5}$ there.

And we know that $5$ goes into $15$.

So, we might as well break it down. So we’ll have

$\sqrt{3} \sqrt{5} \sqrt{9} \sqrt{5}$

Let’s just reorganize this

$\sqrt{9} \sqrt{5} \sqrt{5} \sqrt{3}$

And we’ll solve

$3 \times 5 \sqrt{3}$