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

$\sqrt{4}=2$ 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 $3\sqrt{7}$