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$\frac{1}{4 - \sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+ \frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}=?$
Given :
The given expression is $\frac{1}{4 - \sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+ \frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}$
To do :
We have to find the value of the given expression.
Solution :
$\frac{1}{4 - \sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+ \frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}$
Rationalising the denominator we get,
$\frac{1}{4 - \sqrt{15}} = \frac{1}{\sqrt{16} - \sqrt{15}} = \frac{1}{\sqrt{16} - \sqrt{15}} \times \frac{\sqrt{16} + \sqrt{15}}{\sqrt{16} + \sqrt{15}} = \frac{\sqrt{16} + \sqrt{15}}{(\sqrt{16})^2 - (\sqrt{15})^2} =\frac{\sqrt{16} + \sqrt{15}}{16 -15} = \sqrt{16} + \sqrt{15} $.
$\frac{1}{\sqrt{15} - \sqrt{14}} = \frac{1}{\sqrt{15} - \sqrt{14}} \times \frac{\sqrt{15} + \sqrt{14}}{\sqrt{15} + \sqrt{14}} = \frac{\sqrt{15} + \sqrt{14}}{(\sqrt{15})^2 - (\sqrt{14})^2} =\frac{\sqrt{15} + \sqrt{14}}{15 -14} = \sqrt{15} + \sqrt{14} $.
Similarly,
$\frac{1}{\sqrt{14} - \sqrt{13}} = \sqrt{14} + \sqrt{13} $.
$\frac{1}{\sqrt{13} - \sqrt{12}} = \sqrt{13} + \sqrt{12} $.
$\frac{1}{\sqrt{12} - \sqrt{11}} = \sqrt{12} + \sqrt{11} $.
$\frac{1}{\sqrt{11} - \sqrt{10}} = \sqrt{11} + \sqrt{10} $.
$\frac{1}{\sqrt{10} - \sqrt{9}} = \sqrt{10} + \sqrt{9} $.
Therefore,
$\frac{1}{4 - \sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+ \frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3} \Rightarrow$
$ =\sqrt{16} + \sqrt{15} - (\sqrt{15} + \sqrt{14}) + \sqrt{14} + \sqrt{13} - (\sqrt{13} + \sqrt{12}) + \sqrt{12} + \sqrt{11} - (\sqrt{11} + \sqrt{10}) + \sqrt{10} + \sqrt{19}$
$= 4 + \sqrt{15} - \sqrt{15} - \sqrt{14} + \sqrt{14} + \sqrt{13} - \sqrt{13} + \sqrt{12} - \sqrt{12} + \sqrt{11} - \sqrt{11} - \sqrt{10} + \sqrt{10} + 3$.
$ = 4+3$
$ = 7$.
Therefore, the value of $\frac{1}{4 - \sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+ \frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}$ is 7.
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