The value of $$\frac{1}{{\sqrt 7 - \sqrt 6 }} - $$  $$\frac{1}{{\sqrt 6 - \sqrt 5 }} + $$  $$\frac{1}{{\sqrt 5 - 2 }} - $$  $$\frac{1}{{\sqrt 8 - \sqrt 7 }} + $$  $$\frac{1}{{3 - \sqrt 8 }} = ?$$

The value of $$\frac{1}{{\sqrt 7 - \sqrt 6 }} - $$  $$\frac{1}{{\sqrt 6 - \sqrt 5 }} + $$  $$\frac{1}{{\sqrt 5 - 2 }} - $$  $$\frac{1}{{\sqrt 8 - \sqrt 7 }} + $$  $$\frac{1}{{3 - \sqrt 8 }} = ?$$ Correct Answer 5

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