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\( \displaystyle{1 \over {\sqrt 1 - \sqrt 2 }} - {1 \over {\sqrt 2 - \sqrt 3 }} + {1 \over {\sqrt 3 - \sqrt 4 }}\) \( \displaystyle - {1 \over {\sqrt 4 - \sqrt 5 }} + {1 \over {\sqrt 5 - \sqrt 6 }} -{1 \over {\sqrt 6 - \sqrt 7 }}\) \( \displaystyle + {1 \over {\sqrt 7 - \sqrt 8 }} - {1 \over {\sqrt 8 - \sqrt 9 }}\)
\( \displaystyle{1 \over {\sqrt 1 - \sqrt 2 }} - {1 \over {\sqrt 2 - \sqrt 3 }} + {1 \over {\sqrt 3 - \sqrt 4 }}\) \( \displaystyle - {1 \over {\sqrt 4 - \sqrt 5 }} + {1 \over {\sqrt 5 - \sqrt 6 }} -{1 \over {\sqrt 6 - \sqrt 7 }}\) \( \displaystyle + {1 \over {\sqrt 7 - \sqrt 8 }} - {1 \over {\sqrt 8 - \sqrt 9 }}\)
Phương pháp giải
Áp dụng:\(\dfrac{1}{{\sqrt A - \sqrt B }} \)\(= \dfrac{{\sqrt A + \sqrt B }}{{A - B}} \left( {A,B \ge 0; A \ne B} \right)\)
Lời giải chi tiết
Ta có:
\( \displaystyle{1 \over {\sqrt 1 - \sqrt 2 }} - {1 \over {\sqrt 2 - \sqrt 3 }} + {1 \over {\sqrt 3 - \sqrt 4 }}\) \( \displaystyle- {1 \over {\sqrt 4 - \sqrt 5 }} + {1 \over {\sqrt 5 - \sqrt 6 }} - {1 \over {\sqrt 6 - \sqrt 7 }}\) \( \displaystyle+ {1 \over {\sqrt 7 - \sqrt 8 }} - {1 \over {\sqrt 8 - \sqrt 9 }}\)
\( \displaystyle = {{\sqrt 1 + \sqrt 2 } \over {{{(\sqrt 1 )}^2} - {{(\sqrt 2 )}^2}}} - {{\sqrt 2 + \sqrt 3 } \over {{{(\sqrt 2 )}^2} - {{(\sqrt 3 )}^2}}}\) \( \displaystyle + {{\sqrt 3 + \sqrt 4 } \over {{{(\sqrt 3 )}^2} - {{(\sqrt 4 )}^2}}} - {{\sqrt 4 + \sqrt 5 } \over {{{(\sqrt 4 )}^2} - {{(\sqrt 5 )}^2}}} + \)
\( \displaystyle+ {{\sqrt 5 + \sqrt 6 } \over {{{(\sqrt 5 )}^2} - {{(\sqrt 6 )}^2}}} - {{\sqrt 6 + \sqrt 7 } \over {{{(\sqrt 6 )}^2} - {{(\sqrt 7 )}^2}}}\) \( \displaystyle + {{\sqrt 7 + \sqrt 8 } \over {{{(\sqrt 7 )}^2} - {{(\sqrt 8 )}^2}}} - {{\sqrt 8 + \sqrt 9 } \over {{{(\sqrt 8 )}^2} - {{(\sqrt 9 )}^2}}}\)
\( \displaystyle = {{\sqrt 1 + \sqrt 2 } \over {1 - 2}} - {{\sqrt 2 + \sqrt 3 } \over {2 - 3}} \displaystyle+ {{\sqrt 3 + \sqrt 4 } \over {3 - 4}}\) \( \displaystyle- {{\sqrt 4 + \sqrt 5 } \over {4 - 5}} \displaystyle + {{\sqrt 5 + \sqrt 6 } \over {5 - 6}} - {{\sqrt 6 + \sqrt 7 } \over {6 - 7}}\) \( \displaystyle + {{\sqrt 7 + \sqrt 8 } \over {7 - 8}} - {{\sqrt 8 + \sqrt 9 } \over {8 - 9}}\)
\( \displaystyle= {{\sqrt 1 + \sqrt 2 } \over { - 1}} - {{\sqrt 2 + \sqrt 3 } \over { - 1}} \displaystyle+ {{\sqrt 3 + \sqrt 4 } \over { - 1}}\) \( \displaystyle - {{\sqrt 4 + \sqrt 5 } \over { - 1}} + {{\sqrt 5 + \sqrt 6 } \over { - 1}} - {{\sqrt 6 + \sqrt 7 } \over { - 1}}\) \( \displaystyle+ {{\sqrt 7 + \sqrt 8 } \over { - 1}} - {{\sqrt 8 + \sqrt 9 } \over { - 1}}\)
\( \displaystyle = {{\sqrt 1 - \sqrt 9 } \over { - 1}}\)
\( \displaystyle = \sqrt 9 - \sqrt 1 = 3 - 1 = 2\)
Áp dụng:\(\dfrac{1}{{\sqrt A - \sqrt B }} \)\(= \dfrac{{\sqrt A + \sqrt B }}{{A - B}} \left( {A,B \ge 0; A \ne B} \right)\)
Lời giải chi tiết
Ta có:
\( \displaystyle{1 \over {\sqrt 1 - \sqrt 2 }} - {1 \over {\sqrt 2 - \sqrt 3 }} + {1 \over {\sqrt 3 - \sqrt 4 }}\) \( \displaystyle- {1 \over {\sqrt 4 - \sqrt 5 }} + {1 \over {\sqrt 5 - \sqrt 6 }} - {1 \over {\sqrt 6 - \sqrt 7 }}\) \( \displaystyle+ {1 \over {\sqrt 7 - \sqrt 8 }} - {1 \over {\sqrt 8 - \sqrt 9 }}\)
\( \displaystyle = {{\sqrt 1 + \sqrt 2 } \over {{{(\sqrt 1 )}^2} - {{(\sqrt 2 )}^2}}} - {{\sqrt 2 + \sqrt 3 } \over {{{(\sqrt 2 )}^2} - {{(\sqrt 3 )}^2}}}\) \( \displaystyle + {{\sqrt 3 + \sqrt 4 } \over {{{(\sqrt 3 )}^2} - {{(\sqrt 4 )}^2}}} - {{\sqrt 4 + \sqrt 5 } \over {{{(\sqrt 4 )}^2} - {{(\sqrt 5 )}^2}}} + \)
\( \displaystyle+ {{\sqrt 5 + \sqrt 6 } \over {{{(\sqrt 5 )}^2} - {{(\sqrt 6 )}^2}}} - {{\sqrt 6 + \sqrt 7 } \over {{{(\sqrt 6 )}^2} - {{(\sqrt 7 )}^2}}}\) \( \displaystyle + {{\sqrt 7 + \sqrt 8 } \over {{{(\sqrt 7 )}^2} - {{(\sqrt 8 )}^2}}} - {{\sqrt 8 + \sqrt 9 } \over {{{(\sqrt 8 )}^2} - {{(\sqrt 9 )}^2}}}\)
\( \displaystyle = {{\sqrt 1 + \sqrt 2 } \over {1 - 2}} - {{\sqrt 2 + \sqrt 3 } \over {2 - 3}} \displaystyle+ {{\sqrt 3 + \sqrt 4 } \over {3 - 4}}\) \( \displaystyle- {{\sqrt 4 + \sqrt 5 } \over {4 - 5}} \displaystyle + {{\sqrt 5 + \sqrt 6 } \over {5 - 6}} - {{\sqrt 6 + \sqrt 7 } \over {6 - 7}}\) \( \displaystyle + {{\sqrt 7 + \sqrt 8 } \over {7 - 8}} - {{\sqrt 8 + \sqrt 9 } \over {8 - 9}}\)
\( \displaystyle= {{\sqrt 1 + \sqrt 2 } \over { - 1}} - {{\sqrt 2 + \sqrt 3 } \over { - 1}} \displaystyle+ {{\sqrt 3 + \sqrt 4 } \over { - 1}}\) \( \displaystyle - {{\sqrt 4 + \sqrt 5 } \over { - 1}} + {{\sqrt 5 + \sqrt 6 } \over { - 1}} - {{\sqrt 6 + \sqrt 7 } \over { - 1}}\) \( \displaystyle+ {{\sqrt 7 + \sqrt 8 } \over { - 1}} - {{\sqrt 8 + \sqrt 9 } \over { - 1}}\)
\( \displaystyle = {{\sqrt 1 - \sqrt 9 } \over { - 1}}\)
\( \displaystyle = \sqrt 9 - \sqrt 1 = 3 - 1 = 2\)