Question
$$\frac{1}{{\left( {\sqrt 9 - \sqrt 8 } \right)}} \, - $$ $$\frac{1}{{\left( {\sqrt 8 - \sqrt 7 } \right)}} \, + $$ $$\frac{1}{{\left( {\sqrt 7 - \sqrt 6 } \right)}} \, - $$ $$\frac{1}{{\left( {\sqrt 6 - \sqrt 5 } \right)}} \, + $$ $$\frac{1}{{\left( {\sqrt 5 - \sqrt 4 } \right)}}$$ is equal to ?
Answer: Option D
Given expression,
$$ = \frac{1}{{\left( {\sqrt 9 - \sqrt 8 } \right)}} \times \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{{\left( {\sqrt 9 + \sqrt 8 } \right)}}$$ $$ - \frac{1}{{\left( {\sqrt 8 - \sqrt 7 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{{\left( {\sqrt 8 + \sqrt 7 } \right)}}$$ $$ + \frac{1}{{\left( {\sqrt 7 - \sqrt 6 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{{\left( {\sqrt 7 + \sqrt 6 } \right)}}$$ $$ - \frac{1}{{\left( {\sqrt 6 - \sqrt 5 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {\sqrt 6 + \sqrt 5 } \right)}}$$ $$ + \frac{1}{{\left( {\sqrt 5 - \sqrt 4 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{{\left( {\sqrt 5 + \sqrt 4 } \right)}}$$
$$ = \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{{\left( {9 - 8} \right)}} - \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{{\left( {8 - 7} \right)}}$$ $$ + \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{{\left( {7 - 6} \right)}}$$ $$ - \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {6 - 5} \right)}}$$ $$ + \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{{\left( {5 - 4} \right)}}$$
$$ = \left( {\sqrt 9 + \sqrt 8 } \right) - \left( {\sqrt 8 + \sqrt 7 } \right)$$ $$ + \left( {\sqrt 7 + \sqrt 6 } \right)$$ $$ - \left( {\sqrt 6 + \sqrt 5 } \right)$$ $$ + \left( {\sqrt 5 + \sqrt 4 } \right)$$
$$ = \left( {\sqrt 9 + \sqrt 4 } \right)$$
$$ = 3 + 2$$
$$ = 5$$
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Given expression,
$$ = \frac{1}{{\left( {\sqrt 9 - \sqrt 8 } \right)}} \times \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{{\left( {\sqrt 9 + \sqrt 8 } \right)}}$$ $$ - \frac{1}{{\left( {\sqrt 8 - \sqrt 7 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{{\left( {\sqrt 8 + \sqrt 7 } \right)}}$$ $$ + \frac{1}{{\left( {\sqrt 7 - \sqrt 6 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{{\left( {\sqrt 7 + \sqrt 6 } \right)}}$$ $$ - \frac{1}{{\left( {\sqrt 6 - \sqrt 5 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {\sqrt 6 + \sqrt 5 } \right)}}$$ $$ + \frac{1}{{\left( {\sqrt 5 - \sqrt 4 } \right)}}$$ $$ \times \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{{\left( {\sqrt 5 + \sqrt 4 } \right)}}$$
$$ = \frac{{\left( {\sqrt 9 + \sqrt 8 } \right)}}{{\left( {9 - 8} \right)}} - \frac{{\left( {\sqrt 8 + \sqrt 7 } \right)}}{{\left( {8 - 7} \right)}}$$ $$ + \frac{{\left( {\sqrt 7 + \sqrt 6 } \right)}}{{\left( {7 - 6} \right)}}$$ $$ - \frac{{\left( {\sqrt 6 + \sqrt 5 } \right)}}{{\left( {6 - 5} \right)}}$$ $$ + \frac{{\left( {\sqrt 5 + \sqrt 4 } \right)}}{{\left( {5 - 4} \right)}}$$
$$ = \left( {\sqrt 9 + \sqrt 8 } \right) - \left( {\sqrt 8 + \sqrt 7 } \right)$$ $$ + \left( {\sqrt 7 + \sqrt 6 } \right)$$ $$ - \left( {\sqrt 6 + \sqrt 5 } \right)$$ $$ + \left( {\sqrt 5 + \sqrt 4 } \right)$$
$$ = \left( {\sqrt 9 + \sqrt 4 } \right)$$
$$ = 3 + 2$$
$$ = 5$$
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