a) \(A = \frac{{{x^{\frac{1}{3}}}\sqrt y + {y^{\frac{1}{3}}}\sqrt x }}{{\sqrt[6]{x} + \sqrt[6]{y}}}\)
\(= \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{2}}} + {y^{\frac{1}{3}}}.{x^{\frac{1}{2}}}}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}} \)
\( = \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{6} + \frac{1}{3}}} + {y^{\frac{1}{3}}}.{x^{\frac{1}{6} + \frac{1}{3}}}}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}}\)
\( = \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{6}}}.{y^{\frac{1}{3}}} + {y^{\frac{1}{3}}}.{x^{\frac{1}{6}}}.{x^{\frac{1}{3}}}}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}}\)
\(= \frac{{{x^{\frac{1}{3}}}.{y^{\frac{1}{3}}}\left( {{y^{\frac{1}{6}}} + {x^{\frac{1}{6}}}} \right)}}{{{x^{\frac{1}{6}}} + {y^{\frac{1}{6}}}}}\)
\(= \sqrt[3]{x}.\sqrt[3]{y}\)
\(= \sqrt[3]{{xy}}\).
b) \(B = {\left( {\frac{{{x^{\sqrt 3 }}}}{{{y^{\sqrt 3 - 1}}}}} \right)^{\sqrt 3 + 1}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}}\)
\(= \frac{{{x^{\sqrt 3 .\left( {\sqrt 3 + 1} \right)}}}}{{{y^{\left( {\sqrt 3 - 1} \right)\left( {\sqrt 3 + 1} \right)}}}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}} \)
\(= \frac{{{x^{3 + \sqrt 3 }}}}{{{y^2}}}.\frac{{{x^{ - \left( {\sqrt 3 + 1} \right)}}}}{{{y^{ - 2}}}}\)
\(= \frac{{{x^{3 + \sqrt 3 }}}}{{{y^2}}}.\frac{{{y^2}}}{{{x^{\sqrt 3 + 1}}}}\)
\(= \frac{{{x^{3 + \sqrt 3 }}}}{{{x^{\sqrt 3 + 1}}}} \)
\(= {x^{3 + \sqrt 3 - \sqrt 3 - 1}} \)
\(= {x^2}\).
