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 $15\sqrt{3}$

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 $3 \times 5 \sqrt{3}$