જો $a = \sqrt {(21)} - \sqrt {(20)} $ અને $b = \sqrt {(18)} - \sqrt {(17),} $ તો
Difficult
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d
(d) \(a - b = \sqrt {21} - \sqrt {20} - \sqrt {18} + \sqrt {17} \)
(d) \(a - b = \sqrt {21} - \sqrt {20} - \sqrt {18} + \sqrt {17} \)
= \((\sqrt {21} - \sqrt {18} ) - (\sqrt {20} - \sqrt {17} )\)
= \({{(\sqrt {21} - \sqrt {18} )(\sqrt {21} + \sqrt {18} )} \over {\sqrt {21} + \sqrt {18} }} - {{20 - 17} \over {\sqrt {20} + \sqrt {17} }}\)
= \(3\,\left[ {{1 \over {\sqrt {21} + \sqrt {18} }} - {1 \over {\sqrt {20} + \sqrt {17} }}} \right]\)
= \({{3\,[\sqrt {20} + \sqrt {17} - \sqrt {21} - \sqrt {18} ]} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}}\)
= \({{3\,[(\sqrt {20} - \sqrt {21} ) + (\sqrt {17} - \sqrt {18)} ]} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}}\)
= \({{ - 3\,[(\sqrt {21} - \sqrt {20} ) + (\sqrt {18} - \sqrt {17} )} \over {(\sqrt {21} + \sqrt {18} )\,(\sqrt {20} + \sqrt {17} )}} < 0\),
\(\therefore a < b\).
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