Cho x, y là các số thực dương. Rút gọn các biểu thức sau:
a) \(A = \frac{{{x^{\frac{1}{3}}}\sqrt y + {y^{\frac{1}{3}}}\sqrt x }}{{\sqrt[6]{x} + \sqrt[6]{y}}};\)
b) \(B = {\left( {\frac{{{x^{\sqrt 3 }}}}{{{y^{\sqrt 3 - 1}}}}} \right)^{\sqrt 3 + 1}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}}.\)
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Sử dụng công thức
\({a^{\frac{1}{n}}} = \sqrt[n]{a};{\left( {{a^m}} \right)^n} = {a^{m.n}};{a^m}:{a^n} = {a^{m - n}};{a^{ - n}} = \frac{1}{{{a^n}}}.\)
a) \(A = \frac{{{x^{\frac{1}{3}}}\sqrt y + {y^{\frac{1}{3}}}\sqrt x }}{{\sqrt[6]{x} + \sqrt[6]{y}}} = \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{2}}} + {y^{\frac{1}{3}}}.{x^{\frac{1}{2}}}}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}} = \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{3}}}\left( {{y^{\frac{1}{6}}} + {x^{\frac{1}{6}}}} \right)}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}} = \sqrt[3]{x}.\sqrt[3]{y} = \sqrt[3]{{xy}}\)
b) \(B = {\left( {\frac{{{x^{\sqrt 3 }}}}{{{y^{\sqrt 3 - 1}}}}} \right)^{\sqrt 3 + 1}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}} = \frac{{{x^{\sqrt 3 .\left( {\sqrt 3 + 1} \right)}}}}{{{y^{\left( {\sqrt 3 - 1} \right)\left( {\sqrt 3 + 1} \right)}}}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}} = \frac{{{x^{3 + \sqrt 3 }}}}{{{y^2}}}.\frac{{{x^{ - \left( {\sqrt 3 + 1} \right)}}}}{{{y^{ - 2}}}} = \frac{{{x^{3 + \sqrt 3 }}}}{{{y^2}}}.\frac{{{y^2}}}{{{x^{\sqrt 3 + 1}}}} = \frac{{{x^{3 + \sqrt 3 }}}}{{{x^{\sqrt 3 + 1}}}} = {x^{3 + \sqrt 3 - \sqrt 3 - 1}} = {x^2}\)