# What is sqrt(187200) ?

Feb 3, 2017

$432.666153$

#### Explanation:

The square root of a non-negative number $n$ is the non-negative number which, when squared, gives $n$ as the result.

There isn't really too much to do here. Trying to find a perfect square would be in vain, since the result is a decimal (with six decimal digits, mind you), so your best bet is to just use a calculator.

Feb 3, 2017

$\sqrt{187200} = 120 \sqrt{13}$

#### Explanation:

To simplify $\sqrt{187200}$, first find the prime factorisation of $187200$...

$\textcolor{w h i t e}{0} 187200$
$\textcolor{w h i t e}{00} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0} 2 \textcolor{w h i t e}{000} 93600$
$\textcolor{w h i t e}{00000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000} 2 \textcolor{w h i t e}{000} 46800$
$\textcolor{w h i t e}{00000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000} 2 \textcolor{w h i t e}{000} 23400$
$\textcolor{w h i t e}{00000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000000} 2 \textcolor{w h i t e}{000} 11700$
$\textcolor{w h i t e}{00000000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000000000} 2 \textcolor{w h i t e}{000} 5850$
$\textcolor{w h i t e}{00000000000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000000000000} 2 \textcolor{w h i t e}{000} 2925$
$\textcolor{w h i t e}{00000000000000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000000000000000} 3 \textcolor{w h i t e}{000} 975$
$\textcolor{w h i t e}{00000000000000000000000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{0000000000000000000000} 3 \textcolor{w h i t e}{00} 325$
$\textcolor{w h i t e}{0000000000000000000000000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{000000000000000000000000} 5 \textcolor{w h i t e}{00} 65$
$\textcolor{w h i t e}{00000000000000000000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{0000000000000000000000000} 5 \textcolor{w h i t e}{000} 13$

That is:

$187200 = {2}^{6} \cdot {3}^{2} \cdot {5}^{2} \cdot 13$

So:

$\sqrt{187200} = {2}^{3} \cdot 3 \cdot 5 \sqrt{13} = 120 \sqrt{13}$