The value of \(\left[\frac{1}{\sqrt{9}-\sqrt{8}}\right]-\left[\frac{1}{\sqrt{8}-\sqrt{7}}\right]+\left[\frac{1}{\sqrt{7}-\sqrt{6}}\right]\) \(-\left[\frac{1}{\sqrt{6}-\sqrt{5}}\right]+\left[\frac{1}{\sqrt{5}-\sqrt{4}}\right]\) is:

To simplify the given expression, we can use the conjugate of each denominator to rationalize it. The conjugate of a binomial \(\sqrt{a} – \sqrt{b}\) is \(\sqrt{a} + \sqrt{b}\). Multiplying both the numerator and denominator by the conjugate, we get:

\[
\left[\frac{1}{\sqrt{9}-\sqrt{8}}\right] – \left[\frac{1}{\sqrt{8}-\sqrt{7}}\right] + \left[\frac{1}{\sqrt{7}-\sqrt{6}}\right] – \left[\frac{1}{\sqrt{6}-\sqrt{5}}\right] + \left[\frac{1}{\sqrt{5}-\sqrt{4}}\right]
\]

\[
= \left[\frac{\sqrt{9} + \sqrt{8}}{(\sqrt{9} – \sqrt{8})(\sqrt{9} + \sqrt{8})}\right] – \left[\frac{\sqrt{8} + \sqrt{7}}{(\sqrt{8} – \sqrt{7})(\sqrt{8} + \sqrt{7})}\right] + \left[\frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7} – \sqrt{6})(\sqrt{7} + \sqrt{6})}\right]
\]

\[
– \left[\frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6} – \sqrt{5})(\sqrt{6} + \sqrt{5})}\right] + \left[\frac{\sqrt{5} + \sqrt{4}}{(\sqrt{5} – \sqrt{4})(\sqrt{5} + \sqrt{4})}\right]
\]

\[
= \left[\frac{\sqrt{9} + \sqrt{8}}{9 – 8}\right] – \left[\frac{\sqrt{8} + \sqrt{7}}{8 – 7}\right] + \left[\frac{\sqrt{7} + \sqrt{6}}{7 – 6}\right] – \left[\frac{\sqrt{6} + \sqrt{5}}{6 – 5}\right] + \left[\frac{\sqrt{5} + \sqrt{4}}{5 – 4}\right]
\]

\[
= (\sqrt{9} + \sqrt{8}) – (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) – (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + \sqrt{4})
\]

Now, notice that the terms cancel out in pairs:

\[
= \sqrt{9} + \sqrt{4} = 3 + 2 = 5
\]

Therefore, the value of the given expression is 5.