Rút gọn biểu thức sau: \(\frac{{{a^{\frac{1}{3}}}\sqrt b + {b^{\frac{1}{3}}}\sqrt a }}{{\sqrt[6]{a} + \sqrt[6]{b}}}\).

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

\( = \frac{{{a^{\frac{1}{3}}}\,.\,{b^{\frac{1}{3} + \frac{1}{6}}} + {b^{\frac{1}{3}}}\,.\,{a^{\frac{1}{3} + \frac{1}{6}}}}}{{{a^{\frac{1}{6}}} + {b^{\frac{1}{6}}}}} = \frac{{{a^{\frac{1}{3}}}\,.\,{b^{\frac{1}{3}}}\,.\,{b^{\frac{1}{6}}} + {b^{\frac{1}{3}}}\,.\,{a^{\frac{1}{3}}}\,.\,\,{a^{\frac{1}{6}}}}}{{{a^{\frac{1}{6}}} + {b^{\frac{1}{6}}}}}\)

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