# How do you simplify sqrt(15n^2)*sqrt(10n^3)?

Apr 13, 2017

$5 {n}^{2} \sqrt{6 n}$

#### Explanation:

Demonstrating a principle by example:

$\sqrt{a n} = \sqrt{a} \times \sqrt{n}$

$\sqrt{4 \times 25} = 2 \times 5 = 10$
$\sqrt{4 \times 25} = \sqrt{100} = 10$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:$\text{ } \sqrt{15 {n}^{2}} \times \sqrt{10 {n}^{3}}$

$\text{ } \left[\sqrt{3} \times \sqrt{5} \times \textcolor{b l u e}{\sqrt{{n}^{2}}}\right] \times \left[\sqrt{2} \times \sqrt{5} \times \textcolor{red}{\sqrt{{n}^{2}}} \times \sqrt{n}\right]$

$\leftarrow \text{ "color(blue)(n)[sqrt(3)xxsqrt(5)]" "xx" } \textcolor{red}{n} \left[\sqrt{2} \times \sqrt{5} \times \sqrt{n}\right]$
$| \text{ }$......................................................................................................
$|$
$|$
$\rightarrow \text{ "color(brown)(n)[sqrt(3)xxcolor(blue)(sqrt(5))]" "xx" } \textcolor{b r o w n}{n} \left[\sqrt{2} \times \textcolor{b l u e}{\sqrt{5}} \times \sqrt{n}\right]$

$\text{ } \textcolor{b l u e}{5} \textcolor{b r o w n}{{n}^{2}} \left[\sqrt{3} \times \sqrt{2} \times \sqrt{n}\right]$

$\textcolor{w h i t e}{}$

$\text{ } 5 {n}^{2} \sqrt{6 n}$

Apr 13, 2017

$5 {n}^{2} \sqrt{6 n}$

#### Explanation:

color(blue)(sqrt(15n^2)*sqrt(10n^3)

Split the equation using

color(brown)(sqrt(xy)=sqrtx*sqrty

So,

$\rightarrow \sqrt{15} \cdot \textcolor{red}{\sqrt{{n}^{2}}} \cdot \sqrt{10} \cdot \textcolor{red}{\sqrt{{n}^{2}}} \cdot n$

$\rightarrow \textcolor{red}{n \cdot n} \cdot \underbrace{\sqrt{5}} \cdot \sqrt{3} \cdot \underbrace{\sqrt{5}} \cdot \sqrt{2} \cdot n$

color(green)(rArr5n^2*sqrt(6n)

Hope this helps... :)