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The value of $$\left(\frac{1}{\sqrt{9} - \sqrt{8}} -\frac{1}{\sqrt{8} - \sqrt{7}}+\frac{1}{\sqrt{7} - \sqrt{6}}-\frac{1}{\sqrt{6} - \sqrt{5}}+\frac{1}{\sqrt{5} - \sqrt{4}} \right)$$ is
$$\left(\frac{1}{\sqrt{9} - \sqrt{8}} -\frac{1}{\sqrt{8} - \sqrt{7}}+\frac{1}{\sqrt{7} - \sqrt{6}}-\frac{1}{\sqrt{6} - \sqrt{5}}+\frac{1}{\sqrt{5} - \sqrt{4}} \right)$$
= ($$\sqrt{9\ }+\ \sqrt{\ 8}$$) - ($$\sqrt{8\ }+\ \sqrt{\ 7}$$) + ($$\sqrt{7\ }+\ \sqrt{\ 6}$$) - ($$\sqrt{6\ }+\ \sqrt{\ 5}$$) + ($$\sqrt{5\ }+\ \sqrt{\ 4}$$)
= ($$\sqrt{9\ }+\ \sqrt{\ 4}$$)
= 3 + 2
= 5
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