Rút gọn các biểu thức sau:

a) \(A = \sqrt[3]{{20 + 14\sqrt 2 }} + \sqrt[3]{{20 - 14\sqrt 2 }}\).    b) \(B = \sqrt[3]{{182 + \sqrt {33125} }} + \sqrt[3]{{182 - \sqrt {33125} }}\).

a) Ta có \(A = \left[ {\left( {\frac{1}{a} - \sqrt[6]{{\frac{1}{a}}} + \sqrt[3]{{{a^2}}}} \right) + \left( {\frac{a}{{{a^2}}}\sqrt[6]{{{a^5}}} - \frac{3}{a}\sqrt[3]{{{a^2}}}} \right)} \right]a\sqrt[3]{a}\)

\( = \left[ {\left( {\frac{1}{a} - \sqrt[6]{{\frac{1}{a}}} + \sqrt[3]{{{a^2}}}} \right) + \left( {\sqrt[6]{{\frac{1}{a}}} - 3\sqrt[3]{{\frac{1}{a}}}} \right)} \right]a\sqrt[3]{a}\)

\( = \left( {\frac{1}{a} + \sqrt[3]{{{a^2}}} - 3\sqrt[3]{{\frac{1}{a}}}} \right)a\sqrt[3]{a}\)

\( = {a^2} - 3a + \sqrt[3]{a}\).

b)  Ta có \(B = \sqrt[3]{{{a^2}}}\sqrt {\sqrt[3]{{{a^2}}} + \sqrt[3]{{{b^2}}}}  + \sqrt[3]{{{b^2}}}\sqrt {\sqrt[3]{{{a^2}}} + \sqrt[3]{{{b^2}}}} \)

\( = \left( {\sqrt[3]{{{a^2}}} + \sqrt[3]{{{b^2}}}} \right)\sqrt {\sqrt[3]{{{a^2}}} + \sqrt[3]{{{b^2}}}} \)

\( = \sqrt {{{\left( {\sqrt[3]{{{a^2}}} + \sqrt[3]{{{b^2}}}} \right)}^3}} .\)

Ta có \[x = \frac{2}{{2\sqrt[3]{2} + 2 + \sqrt[3]{4}}} = \frac{2}{{\sqrt[3]{{16}} + 2 + \sqrt[3]{4}}}\]

\[ = \frac{2}{{{{\left( {\sqrt[3]{4}} \right)}^2} + \sqrt[3]{2}.\sqrt[3]{4} + {{\left( {\sqrt[3]{2}} \right)}^2}}}\]

\[ = \frac{{2\left( {\sqrt[3]{4} - \sqrt[3]{2}} \right)}}{{4 - 2}} = \sqrt[3]{4} - \sqrt[3]{2}\]

Tương tự \(y = \frac{6}{{2\sqrt[3]{2} - 2 + \sqrt[3]{4}}} = \sqrt[3]{4} + \sqrt[3]{2}\)

Do đó: \(x{y^3} - {x^3}y = xy\left( {{y^2} - {x^2}} \right) = xy\left( {y - x} \right)\left( {y + x} \right) = 8\left( {2\sqrt[3]{2} - \sqrt[3]{4}} \right)\)