Question

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

Answer 1

`5n^2sqrt(6n)`

Explanation:

Demonstrating a principle by example:

`sqrt(an)=sqrt(a)xxsqrt(n)`

`sqrt(4xx25)=2xx5=10`
`sqrt(4xx25)=sqrt(100)=10`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:`" "sqrt(15n^2)xxsqrt(10n^3)`

`" "[sqrt(3)xxsqrt(5)xxcolor(blue)(sqrt(n^2))]xx[sqrt(2)xxsqrt(5)xxcolor(red)(sqrt(n^2))xxsqrt(n)]`

`larr" "color(blue)(n)[sqrt(3)xxsqrt(5)]" "xx" "color(red)(n)[sqrt(2) xxsqrt(5)xxsqrt(n)]`
`|" "`......................................................................................................
`|`
`|`
`rarr" "color(brown)(n)[sqrt(3)xxcolor(blue)(sqrt(5))]" "xx" "color(brown)(n)[sqrt(2) xxcolor(blue)(sqrt(5))xxsqrt(n)]`

`" "color(blue)(5)color(brown)(n^2)[sqrt(3)xxsqrt(2)xxsqrt(n)]`

`color(white)()`

`" "5n^2sqrt(6n)`

Answer 2

`5n^2sqrt(6n)`

Explanation:

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

Split the equation using

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

So,

`rarrsqrt15*color(red)(sqrt(n^2))*sqrt(10)*color(red)(sqrt(n^2))*n`

`rarrcolor(red)(n*n)*underbracesqrt5*sqrt3*underbracesqrt5*sqrt2*n`

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

Hope this helps... :)