<s>[INST] What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually? What is the fastest method to find which number is bigger manually?
$\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $
 [/INST] $$\dfrac{3\sqrt3-8}{1-2\sqrt3}=\dfrac{8-3\sqrt3}{2\sqrt3-1}$$
Now $$\dfrac{3\sqrt3-4}{7-2\sqrt3}-\dfrac{8-3\sqrt3}{2\sqrt3-1}=\dfrac{(3\sqrt3-4)(2\sqrt3-1)-(7-2\sqrt3)(8-3\sqrt3)}{(7-2\sqrt3)(2\sqrt3-1)}$$
$$=\dfrac{26\sqrt3-52}{(7-2\sqrt3)(2\sqrt3-1)}$$
Now  the denominator $>0,$ and $26\sqrt3-52=26(\sqrt3-2)<0$
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