In this video, we are going to divide radical expressions.

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One simple example is:
$\frac{\sqrt{25}}{\sqrt{9}}$
By simplifying the numerator and the denominator, we now have $\frac{5}{3}$

A slightly more difficult problem would be:
$\frac{\sqrt{50}}{\sqrt{18}}$
Just like the other problem, first find perfect square factors for each expression
$\frac{\sqrt{25}\times\sqrt{2}}{\sqrt{9}\times\sqrt{2}}$
Simplify each expression
$\frac{5\sqrt{2}}{3\sqrt{2}}$
And the final answer is $\frac{5}{3}$

If the expression has a radical in the denominator and a rational number in the numerator, then it is necessary to rationalize the fraction.
For example:
$\frac{3}{\sqrt{10}}$
Multiple the denominator to both the numerator and the denominator. Like terms cancel each other out so the final answer is:
$\frac{3\times\sqrt{10}}{\sqrt{10}\times\sqrt{10}}$
$\frac{3\sqrt{10}}{10}$

## Video-Lesson Transcript

Let’s go over how to divide radical expressions.

Let’s say we have

$\dfrac{\sqrt{25}}{\sqrt{9}}$
$\dfrac{5}{3}$

And there’s not much to do.

But what if we have

$\dfrac{\sqrt{50}}{\sqrt{18}}$
$= \dfrac{\sqrt{25} \sqrt{2}}{\sqrt{9} \sqrt {2}}$
$= \dfrac{5 \sqrt{2}}{3 \sqrt{2}}$

And just like if we have

$\dfrac{5x}{3x}$

$x$ will be cancelled out.

In this case, $\sqrt{2}$ will be cancelled out.

$= \dfrac{5}{3}$

Of course, not all is going to work out like this.

Let’s look at another example.

$\dfrac{\sqrt{27}}{\sqrt{30}}$

Now, let’s break this down.

$= \dfrac{\sqrt{9} \sqrt{3}}{\sqrt{30}}$
$= \dfrac{3 \sqrt{3}}{\sqrt{30}}$

This is how far we can go as far as the radicals are concerned.

But this is not our final answer.

Because we don’t want a radical in the denominator.

But even more so, we can reduce this.

Just like this example:

$\dfrac{\sqrt{15}}{\sqrt{5}}$
$= \dfrac{\sqrt{3}}{\sqrt{1}}$
$= \sqrt{3}$

Another way to look at that is:

$\dfrac{\sqrt{15}}{\sqrt{5}}$
$= \dfrac{\sqrt{5} \sqrt{3}}{\sqrt{5}}$

$\sqrt{5}$ cancels out and we’re left with

$= \sqrt{3}$

Now, let’s apply this in solving. Let’s go back

$= \dfrac{3 \sqrt{3}}{\sqrt{30}}$

We can break this down even more into

$= \dfrac{3 \sqrt{3}}{\sqrt{3} \sqrt{10}}$

Here, $= \sqrt{3}$ is going to cancel out.

And we have

$= \dfrac{3}{\sqrt{10}}$

Now, we want to rationalize the denominator.

The trick in getting rid of $\sqrt{10}$ is to multiply the numerator and denominator by the same number.

$= \dfrac{3 \times \sqrt{10}}{\sqrt{10} \times \sqrt{10}}$

$= \dfrac{3 \sqrt{10}}{10}$

Now, let’s look at some more.

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

Let’s break this down

$= \dfrac{\sqrt{9} \sqrt{5}}{\sqrt{15}}$
$= \dfrac{3 \sqrt{5}}{\sqrt{15}}$

Then let’s reduce $5$ and $15$

We’ll have

$= \dfrac{3 \sqrt{1}}{\sqrt{3}}$
$= \dfrac{3}{\sqrt{3}}$

Now, we don’t want a radical in the denominator.

We have to rationalize it. So let’s multiply the numerator and denominator by $\sqrt{3}$.

$= \dfrac{3 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$
$= \dfrac{3 \sqrt{3}}{3}$

Which we cancel and we’re left with our final answer

$= \sqrt{3}$

This is a $100\%$ correct way to solve it.

Any way you solve for the correct answer is a good way to do it.

Let me show you another way.

We can reduce this right off the bat.

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

$15$ goes into $45$, so this can be reduced to

$= \dfrac{ \sqrt{3}}{\sqrt{1}}$
$\sqrt{3}$

Two different options to solve the same problem. We have the same answer both ways.