<s>[INST] which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$ Which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$?
Squaring both sides will give me something but I could not go any further.
 [/INST] $\sqrt{7} - \sqrt{6} = (\sqrt7 - \sqrt6 )\frac{\sqrt7 + \sqrt 6}{\sqrt7 + \sqrt 6} = \frac1{\sqrt7 + \sqrt 6} $
Similarily, 
$\sqrt6 - \sqrt 5 = \frac1{\sqrt6 + \sqrt 5}$
Since $ \sqrt 7 > \sqrt 5$,
$$\sqrt7 + \sqrt6 > \sqrt6 + \sqrt5$$ 
$$\implies \frac1{\sqrt7 + \sqrt6} < \frac1{ \sqrt6 + \sqrt5}$$
$$\implies \sqrt{7} - \sqrt{6} < \sqrt{6} - \sqrt{5}$$
So $ \sqrt6 - \sqrt5$ is the greater one.
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