simplifying radicals

are u sure this can be simplified?

thats the simplest form i can think of

Simplify

$\sqrt{15-6\sqrt6-10\sqrt2+8\sqrt3}$

$\begin{array}{|rcll|} \hline 15 &=& 2+3+4+6 \\ 15 &=& (\sqrt2)^2+(\sqrt3)^2+(\sqrt4)^2+(\sqrt6)^2 \\ &&\boxed{ \text{Let }a=\sqrt2 \\ \text{Let }b=\sqrt3 \\ \text{Let }c=\sqrt4 \\ \text{Let }d=\sqrt6 }\\15 &=& a^2+b^2+c^2+d^2 \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline (a-b-c+d)^2 &=& a^2+b^2+c^2+d^2-2ab-2ac+2ad+2bc-2bd-2cd \\ &&\boxed{ 2ab =2\sqrt6 \\ 2ac = 4\sqrt2 \\ 2ad = 4\sqrt3 \\ 2bc = 4\sqrt3 \\ 2bd = 6\sqrt2 \\ 2cd = 4\sqrt6 } \\ ( \sqrt2 - \sqrt3-\sqrt4+\sqrt6)^2 &=& a^2+b^2+c^2+d^2-2\sqrt6 - 4\sqrt2 + 4\sqrt3 + 4\sqrt3 -6\sqrt2 - 4\sqrt6 \\ ( \sqrt2 - \sqrt3-\sqrt4+\sqrt6)^2 &=& 15-6\sqrt6 - 10\sqrt2 + 8\sqrt3 \\ ( -2+\sqrt2 - \sqrt3+\sqrt6)^2 &=& 15-6\sqrt6 - 10\sqrt2 + 8\sqrt3 \\ \mathbf{-2+\sqrt2 - \sqrt3+\sqrt6} &=& \mathbf{ \sqrt{15-6\sqrt6 - 10\sqrt2 + 8\sqrt3} } \\ \hline \end{array}$