<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] The solution below is a lot of work to do by hand but at least it does not need any bright ideas for algebraic manipulation.
The problem reduces to finding $\dfrac{1}{\sqrt7+2\sqrt5-\sqrt3}$.
Now, $\theta=\sqrt7+2\sqrt5-\sqrt3$ is a root of $x^8 - 120 x^6 + 3632 x^4 - 28800 x^2 + 256$. The extended Euclidean algorithm gives
$$
1 = \frac{1}{256}(x^8 - 120 x^6 + 3632 x^4 - 28800 x^2 + 256)+ \frac{1}{256}(-x^7 + 120 x^5 - 3632 x^3 + 28800 x)x
$$
Therefore,
$$
\dfrac{1}{\sqrt7+2\sqrt5-\sqrt3} = \frac{1}{\theta} =\frac{1}{256}(-\theta^7 + 120 \theta^5 - 3632 \theta^3 + 28800 \theta)
$$
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