The value of $$\frac{1}{{\sqrt 7 - \sqrt 6 }} - \frac{1}{{\sqrt 6 - \sqrt 5 }} + \frac{1}{{\sqrt 5 - 2}} - \frac{1}{{\sqrt 8 - \sqrt 7 }} + \frac{1}{{3 - \sqrt 8 }}{\text{is:}}$$

$$\eqalign{ & \frac{1}{{\sqrt 7 - \sqrt 6 }} - \frac{1}{{\sqrt 6 - \sqrt 5 }} + \frac{1}{{\sqrt 5 - 2}} - \frac{1}{{\sqrt 8 - \sqrt 7 }} + \frac{1}{{3 - \sqrt 8 }} \cr & \Rightarrow {\text{Rationalising}} \cr & \Rightarrow \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{{\left( {\sqrt 7 - \sqrt 6 } \right)\left( {\sqrt 7 + \sqrt 6 } \right)}} - \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {\sqrt 6 - \sqrt 5 } \right)\left( {\sqrt 6 + \sqrt 5 } \right)}} + \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{{\left( {\sqrt 5 - \sqrt 4 } \right)\left( {\sqrt 5 + \sqrt 4 } \right)}} - \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{{\left( {\sqrt 8 - \sqrt 7 } \right)\left( {\sqrt 8 + \sqrt 7 } \right)}} + \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{{\left( {\sqrt 9 - \sqrt 8 } \right)\left( {\sqrt 9 + \sqrt 8 } \right)}} \cr & \Rightarrow \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{1} - \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{1} + \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{1} - \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{1} + \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{1} \cr & \Rightarrow \sqrt 7 + \sqrt 6 - \sqrt 6 - \sqrt 5 + \sqrt 5 + \sqrt 4 - \sqrt 8 - \sqrt 7 + \sqrt 9 + \sqrt 8 \cr & \Rightarrow \sqrt 4 + \sqrt 9 \cr & \Rightarrow 2 + 3 \cr & \Rightarrow 5 \cr} $$