Đề bài
Rút gọn :
a] \[\left[ {\dfrac{y}{{\sqrt y }} - \dfrac{{\sqrt y }}{{\sqrt y + 1}}} \right]:\dfrac{{\sqrt y }}{{y + \sqrt y }}\] với \[y > 0\];
b] \[\left[ {\dfrac{{\sqrt a }}{{\sqrt a - 1}} - \dfrac{1}{{a - \sqrt a }}} \right]:\left[ {\dfrac{1}{{\sqrt a + 1}} + \dfrac{2}{{a - 1}}} \right]\] với \[a > 0,a \ne 1\];
c] \[\left[ {\dfrac{{x\sqrt x + 1}}{{x\sqrt x + x + \sqrt x + 1}} - \dfrac{{\sqrt x }}{{x + 1}}} \right]:\dfrac{{\sqrt x - 1}}{{x + 1}}\] với \[x \ge 0,x \ne 1\];
d] \[\left[ {\dfrac{{\sqrt x - \sqrt y }}{{1 + \sqrt {xy} }} + \dfrac{{\sqrt x + \sqrt y }}{{1 - \sqrt {xy} }}} \right]:\left[ {\dfrac{{x + y + 2xy}}{{1 - xy}} + 1} \right]\]với \[x \ge 0,y \ge 0,xy \ne 1\].
Phương pháp giải - Xem chi tiết
+] Quy đồng mẫu các phân thức.
+] Biến đổi và rút gọn biểu thức.
Lời giải chi tiết
\[\begin{array}{l}a]\;\;\left[ {\dfrac{y}{{\sqrt y }} - \dfrac{{\sqrt y }}{{\sqrt y + 1}}} \right]:\dfrac{{\sqrt y }}{{y + \sqrt y }}\;\;\;\left[ {y > 0} \right]\\ = \left[ {\sqrt y - \dfrac{{\sqrt y }}{{\sqrt y + 1}}} \right]:\dfrac{{\sqrt y }}{{\sqrt y \left[ {\sqrt y + 1} \right]}}\\ = \dfrac{{\sqrt y \left[ {\sqrt y + 1} \right] - \sqrt y }}{{\sqrt y + 1}}:\dfrac{1}{{\sqrt y + 1}}\\ = \dfrac{{y + \sqrt y - \sqrt y }}{{\sqrt y + 1}}.\left[ {\sqrt y + 1} \right]\\ = y.\end{array}\]
\[\begin{array}{l}b]\;\left[ {\dfrac{{\sqrt a }}{{\sqrt a - 1}} - \dfrac{1}{{a - \sqrt a }}} \right]:\left[ {\dfrac{1}{{\sqrt a + 1}} + \dfrac{2}{{a - 1}}} \right]\;\;\;\left[ {a > 0,\;\;a \ne 1} \right]\\ = \left[ {\dfrac{{\sqrt a }}{{\sqrt a - 1}} - \dfrac{1}{{\sqrt a \left[ {\sqrt a - 1} \right]}}} \right]:\left[ {\dfrac{1}{{\sqrt a + 1}} + \dfrac{2}{{\left[ {\sqrt a - 1} \right]\left[ {\sqrt a + 1} \right]}}} \right]\\ = \dfrac{{a - 1}}{{\sqrt a \left[ {\sqrt a - 1} \right]}}:\dfrac{{\sqrt a - 1 + 2}}{{\left[ {\sqrt a - 1} \right]\left[ {\sqrt a + 1} \right]}}\\ = \dfrac{{\left[ {\sqrt a - 1} \right]\left[ {\sqrt a + 1} \right]}}{{\sqrt a \left[ {\sqrt a - 1} \right]}}.\dfrac{{\left[ {\sqrt a - 1} \right]\left[ {\sqrt a + 1} \right]}}{{\sqrt a + 1}}\\ = \dfrac{{\sqrt a + 1}}{{\sqrt a }}.\dfrac{{\sqrt {a - 1} }}{1} = \dfrac{{a - 1}}{{\sqrt a }}.\end{array}\]
\[\begin{array}{l}c]\;\left[ {\dfrac{{x\sqrt x + 1}}{{x\sqrt x + x + \sqrt x + 1}} - \dfrac{{\sqrt x }}{{x + 1}}} \right]:\dfrac{{\sqrt x - 1}}{{x + 1}}\;\;\left[ {x \ge 0;\;\;x \ne 1} \right]\\ = \left[ {\dfrac{{\left[ {\sqrt x + 1} \right]\left[ {x - \sqrt x + 1} \right]}}{{x\left[ {\sqrt x + 1} \right] + \left[ {\sqrt x + 1} \right]}} - \dfrac{{\sqrt x }}{{x + 1}}} \right].\dfrac{{x + 1}}{{\sqrt x - 1}}\\ = \left[ {\dfrac{{\left[ {\sqrt x + 1} \right]\left[ {x - \sqrt x + 1} \right]}}{{\left[ {\sqrt x + 1} \right]\left[ {x + 1} \right]}} - \dfrac{{\sqrt x }}{{x + 1}}} \right].\dfrac{{x + 1}}{{\sqrt x - 1}}\\ = \left[ {\dfrac{{x - \sqrt x + 1}}{{x + 1}} - \dfrac{{\sqrt x }}{{x + 1}}} \right].\dfrac{{x + 1}}{{\sqrt x - 1}}\\ = \dfrac{{x - \sqrt x + 1 - \sqrt x }}{{x + 1}}.\dfrac{{x + 1}}{{\sqrt x - 1}}\\ = \dfrac{{{{\left[ {\sqrt x - 1} \right]}^2}}}{{\sqrt x - 1}} = \sqrt x - 1.\end{array}\]
\[\begin{array}{l}d]\;\left[ {\dfrac{{\sqrt x - \sqrt y }}{{1 + \sqrt {xy} }} + \dfrac{{\sqrt x + \sqrt y }}{{1 - \sqrt {xy} }}} \right]:\left[ {\dfrac{{x + y + 2xy}}{{1 - xy}} + 1} \right]\;\;\left[ {x \ge 0,\;y \ge 0,\;xy \ne 1} \right]\\ = \dfrac{{\left[ {\sqrt x + \sqrt y } \right]\left[ {1 - \sqrt {xy} } \right] + \left[ {\sqrt x + \sqrt y } \right]\left[ {1 + \sqrt {xy} } \right]}}{{\left[ {1 - \sqrt {xy} } \right]\left[ {1 + \sqrt {xy} } \right]}}:\dfrac{{x + y + 2xy + 1 - xy}}{{1 - xy}}\\ = \dfrac{{\sqrt x - x\sqrt y + \sqrt y - y\sqrt x + \sqrt x + x\sqrt y + \sqrt y + y\sqrt x }}{{1 - xy}}.\dfrac{{1 - xy}}{{x + y + xy + 1}}\\ = \dfrac{{2\sqrt x + 2\sqrt y }}{{x + y + xy + 1}} = \dfrac{{2\left[ {\sqrt x + \sqrt y } \right]}}{{x + 1 + y\left[ {x + 1} \right]}}\\ = \dfrac{{2\left[ {\sqrt x + \sqrt y } \right]}}{{\left[ {x + 1} \right]\left[ {y + 1} \right]}}.\end{array}\]