# How do you find the square root of 320?

Sep 20, 2015

$8 \cdot \sqrt{5}$

#### Explanation:

I don't know either,
so let's break it down into pieces, shall we?

We have:
$\sqrt{320}$
The only thing that involves 32 in my mind is $4 \cdot 8$ or $2 \cdot 16$, and so we notice that $320 = 2 \cdot 160$ or $4 \cdot 80$ or $16 \cdot 20$ or $8 \cdot 40$, etc...

Let's try with $4 \cdot 80$:

$\sqrt{320} = \sqrt{4 \cdot 80}$
At this point, it is good to remember the rule:
$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$
so that
$\sqrt{320} = \sqrt{4 \cdot 80}$
$= \sqrt{4} \cdot \sqrt{80}$
$= 2 \cdot \sqrt{80}$

Then you see that $80 = 4 \cdot 20$, so:

$\sqrt{320} = 2 \cdot \sqrt{80}$
$= 2 \cdot \sqrt{4 \cdot 20}$
$= 2 \cdot \sqrt{4} \cdot \sqrt{20}$
$= 2 \cdot 2 \cdot \sqrt{20} = 4 \cdot \sqrt{20}$

Again you see that $20 = 4 \cdot 5$, so:
$\sqrt{320} = 4 \cdot \sqrt{20}$
$= 4 \cdot \sqrt{4 \cdot 5}$
$= 4 \cdot \sqrt{4} \cdot \sqrt{5}$
$= 4 \cdot 2 \cdot \sqrt{5}$
$= 8 \cdot \sqrt{5}$