Bài tập 2.3 trang 100 SBT Toán 12

a) \(\frac{{{a^{\frac{4}{3}}}\left( {{a^{ - \frac{1}{3}}} + {a^{\frac{2}{3}}}} \right)}}{{{a^{\frac{1}{4}}}\left( {{a^{\frac{3}{4}}} + {a^{ - \frac{1}{4}}}} \right)}} = \frac{{{a^{\frac{3}{4}}}.{a^{ - \frac{1}{3}}} + {a^{\frac{4}{3}}}.{a^{\frac{2}{3}}}}}{{{a^{\frac{1}{4}}}.{a^{\frac{3}{4}}} + {a^{\frac{1}{4}}}.{a^{\frac{{ - 1}}{4}}}}} = \frac{{{a^{\frac{4}{3} + \frac{{ - 1}}{3}}} + {a^{\frac{4}{3} + \frac{2}{3}}}}}{{{a^{\frac{1}{4} + \frac{3}{4}}} + {a^{\frac{1}{4} + \frac{{ - 1}}{4}}}}} = \frac{{a + {a^2}}}{{a + 1}} = \frac{{a\left( {1 + a} \right)}}{{1 + a}} = a\)

b) \(\frac{{{a^{\frac{1}{3}}}\sqrt b  + {b^{\frac{1}{3}}}\sqrt a }}{{\sqrt[6]{a} + \sqrt[6]{b}}} = \frac{{{a^{\frac{1}{3}}}.{b^{\frac{1}{2}}} + {b^{\frac{1}{3}}}.{a^{\frac{1}{2}}}}}{{{a^{\frac{1}{6}}} + {b^{\frac{1}{6}}}}} = {a^{\frac{1}{3}}}.{b^{\frac{1}{3}}} = \sqrt[3]{{ab}}\)

c) \(\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)\left( {{a^{\frac{2}{3}}} + {b^{\frac{2}{3}}} - \sqrt[3]{{ab}}} \right)\)

\( = {\left( {{a^{\frac{1}{3}}}} \right)^3} + {\left( {{b^{\frac{1}{3}}}} \right)^3} = a + b\)

d) \(\left( {{a^{\frac{1}{3}}} + {b^{\frac{1}{3}}}} \right):\left( {2 + \sqrt[3]{{\frac{a}{b}}} + \sqrt[3]{{\frac{b}{a}}}} \right) = \frac{{{a^{\frac{1}{3}}} + {b^{\frac{1}{3}}}}}{{2{a^{\frac{1}{3}}}.{b^{\frac{1}{3}}} + {a^{\frac{2}{3}}} + {b^{\frac{2}{3}}}}} = \frac{{\sqrt[3]{{ab}}\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)}}{{{{\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)}^2}}} = \frac{{\sqrt[3]{{ab}}}}{{\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)}}\)

-- Mod Toán 12 HỌC247