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Rút gọn:
\({1 \over {\sqrt 1 – \sqrt 2 }} – {1 \over {\sqrt 2 – \sqrt 3 }} + {1 \over {\sqrt 3 – \sqrt 4 }} – {1 \over {\sqrt 4 – \sqrt 5 }} + {1 \over {\sqrt 5 – \sqrt 6 }} – \)
\(-{1 \over {\sqrt 6 – \sqrt 7 }} + {1 \over {\sqrt 7 – \sqrt 8 }} – {1 \over {\sqrt 8 – \sqrt 9 }}\)
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Ta có:
\({1 \over {\sqrt 1 – \sqrt 2 }} – {1 \over {\sqrt 2 – \sqrt 3 }} + {1 \over {\sqrt 3 – \sqrt 4 }} – {1 \over {\sqrt 4 – \sqrt 5 }} + {1 \over {\sqrt 5 – \sqrt 6 }} -\)
\( – {1 \over {\sqrt 6 – \sqrt 7 }} + {1 \over {\sqrt 7 – \sqrt 8 }} – {1 \over {\sqrt 8 – \sqrt 9 }}\)
\( = {{\sqrt 1 + \sqrt 2 } \over {{{(\sqrt 1 )}^2} – {{(\sqrt 2 )}^2}}} – {{\sqrt 2 + \sqrt 3 } \over {{{(\sqrt 2 )}^2} – {{(\sqrt 3 )}^2}}} + {{\sqrt 3 + \sqrt 4 } \over {{{(\sqrt 3 )}^2} – {{(\sqrt 4 )}^2}}} – {{\sqrt 4 + \sqrt 5 } \over {{{(\sqrt 4 )}^2} – {{(\sqrt 5 )}^2}}} + \)
\(+ {{\sqrt 5 + \sqrt 6 } \over {{{(\sqrt 5 )}^2} – {{(\sqrt 6 )}^2}}} – {{\sqrt 6 + \sqrt 7 } \over {{{(\sqrt 6 )}^2} – {{(\sqrt 7 )}^2}}} + {{\sqrt 7 + \sqrt 8 } \over {{{(\sqrt 7 )}^2} – {{(\sqrt 8 )}^2}}} – {{\sqrt 8 – \sqrt 9 } \over {{{(\sqrt 8 )}^2} – {{(\sqrt 9 )}^2}}}\)
\( = {{\sqrt 1 + \sqrt 2 } \over {1 – 2}} – {{\sqrt 2 + \sqrt 3 } \over {2 – 3}} + {{\sqrt 3 + \sqrt 4 } \over {3 – 4}} – {{\sqrt 4 + \sqrt 5 } \over {4 – 5}} + \)
\( + {{\sqrt 5 + \sqrt 6 } \over {5 – 6}} – {{\sqrt 6 + \sqrt 7 } \over {6 – 7}} + {{\sqrt 7 + \sqrt 8 } \over {7 – 8}} – {{\sqrt 8 – \sqrt 9 } \over {8 – 9}}\)
\(= {{\sqrt 1 + \sqrt 2 } \over { – 1}} – {{\sqrt 2 + \sqrt 3 } \over { – 1}} + {{\sqrt 3 + \sqrt 4 } \over { – 1}} – {{\sqrt 4 + \sqrt 5 } \over { – 1}} + \)
\( + {{\sqrt 5 + \sqrt 6 } \over { – 1}} – {{\sqrt 6 + \sqrt 7 } \over { – 1}} + {{\sqrt 7 + \sqrt 8 } \over { – 1}} – {{\sqrt 8 – \sqrt 9 } \over { – 1}}\)
\( = {{\sqrt 1 – \sqrt 9 } \over { – 1}}\) \( = {{\sqrt 1 – \sqrt 9 } \over { – 1}}\)
\( = \sqrt 9 – \sqrt 1 = 3 – 1 = 2\)