Answer
$-80m^2n^2\sqrt{3m}+7mn^2\sqrt{3mn}$.
Work Step by Step
The given expression is
$=5m^2n\sqrt{12mn^2}+7mn^2\sqrt{3mn}-10\sqrt{243m^5n^4}$
Factor each radicand as perfect square.
$=5m^2n\sqrt{4\cdot 3mn^2}+7mn^2\sqrt{3mn}-10\sqrt{81\cdot 3m^4\cdot mn^4}$
Group the perfect square.
$=5m^2n\sqrt{(4n^2)(3m)}+7mn^2\sqrt{3mn}-10\sqrt{(81m^4n^4)( 3 m)}$
Use product rule.
$=5m^2n\sqrt{4n^2}\sqrt{3m}+7mn^2\sqrt{3mn}-10\sqrt{81m^4n^4}\sqrt{ 3 m}$
Take square root.
$=5m^2n\cdot 2n\sqrt{3m}+7mn^2\sqrt{3mn}-10\cdot 9m^2n^2\sqrt{ 3 m}$
Simplify.
$=10m^2n^2\sqrt{3m}+7mn^2\sqrt{3mn}- 90m^2n^2\sqrt{ 3 m}$
Factor out $m^2n^2\sqrt {3m}$ from the first and the last term.
$=(10-90)m^2n^2\sqrt{3m}+7mn^2\sqrt{3mn}$
Simplify.
$=-80m^2n^2\sqrt{3m}+7mn^2\sqrt{3mn}$.
