<s>[INST] Simplification of $\frac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$ 
Simplify
$$\dfrac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$$
Final solution should have rational denominators.

Suppose the solution is $X$, I have tried to make up an equation for $X^2$
$$X^2 = \frac{200}{-2\sqrt{15}+2\sqrt{35}-\sqrt{21}+10}$$
My idea is that solving for $X^2$, if can be simply done, can easily give us $X$.
Please help!
 [/INST] For a shortcut, let $\,a=\sqrt{3}, b=\sqrt{5}, c=\sqrt{7}\,$, then:
$$
\require{cancel}
\begin{align}
& \dfrac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt{7}+2\sqrt{5}-\sqrt{3}} = \frac{2ac-bc+5ab-16}{c+2b-a}
\\ &\quad\quad\quad\quad =\frac{c(2a-b)\color{red}{+(2b-a)(2a-b)-(2b-a)(2a-b)}+5ab-16}{c+2b-a}
\\ &\quad\quad\quad\quad = \frac{(c+2b-a)(2a-b)-\cancel{5ab}+2a^2+2b^2+\cancel{5ab}-16}{c+2b-a}
\\ &\quad\quad\quad\quad = 2a-b+\frac{\cancel{6 + 10 - 16}}{c+2b-a}
\end{align}
$$
</s>