Đề bài

Cho \(a\) và \(b\) là hai số dương, \(a \ne b\). Rút gọn biểu thức sau: \(A = \left[ {\frac{{a - b}}{{{a^{\frac{3}{4}}} + {a^{\frac{1}{2}}}{b^{\frac{1}{4}}}}} - \frac{{{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}}}{{{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}}}} \right]:\left( {{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}} \right).\)

Lời giải chi tiết

\(\frac{{a - b}}{{{a^{\frac{3}{4}}} + {a^{\frac{1}{2}}}{b^{\frac{1}{4}}}}} = \frac{{a - b}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}}\)\( \Rightarrow B = \frac{{a - b}}{{{a^{\frac{3}{4}}} + {a^{\frac{1}{2}}}{b^{\frac{1}{4}}}}} - \frac{{{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}}}{{{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}}} = \frac{{a - b}}{{{a^{\frac{3}{4}}} + {a^{\frac{1}{2}}}{b^{\frac{1}{4}}}}} - \frac{{{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}}}{{{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}}} = \frac{{a - b}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}} - \frac{{{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}}}{{{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}}}\)\( = \frac{{a - b - {a^{\frac{1}{2}}}\left( {{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}} \right)}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}} = \frac{{{a^{\frac{1}{2}}}{b^{\frac{1}{2}}} - b}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}} = \frac{{{b^{\frac{1}{2}}}\left( {{a^{\frac{1}{2}}} - {b^{\frac{1}{2}}}} \right)}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}}.\) Tacó: \({a^{\frac{1}{2}}} - {b^{\frac{1}{2}}} = \left( {{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}} \right)\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)\)nên \(B = \frac{{{b^{\frac{1}{2}}} \cdot \left( {{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}} \right)\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}}{{{a^{\frac{1}{2}}}\left( {{a^{\frac{1}{4}}} + {b^{\frac{1}{4}}}} \right)}} = {\left( {\frac{b}{a}} \right)^{\frac{1}{2}}} \cdot \left( {{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}} \right).\)Do đó\(A = {\left( {\frac{b}{a}} \right)^{\frac{1}{2}}} \cdot \left( {{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}} \right) \cdot \frac{1}{{{a^{\frac{1}{4}}} - {b^{\frac{1}{4}}}}} = {\left( {\frac{b}{a}} \right)^{\frac{1}{2}}}\)