$$\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{3-\sqrt{8}}$$ is
$$\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{3-\sqrt{8}}$$
Rationalizing each term, we get, the denominator of each term will be 1, we get :
= $$\sqrt{7}$$ + $$\sqrt{6}$$ - ($$\sqrt{6}$$ + $$\sqrt{5}$$) + $$\sqrt{5}$$ + 2 - ($$\sqrt{8}$$ + $$\sqrt{7}$$) + 3 + $$\sqrt{8}$$
= 2+3 = 5
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