Rút gon các biểu thức sau:

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

b) \(B = \sqrt {{a^2} + \sqrt[3]{{{a^4}{b^2}}}}  + \sqrt {{b^2} + \sqrt[3]{{{a^2}{b^4}}}} \).

Rút gọn các biểu thức sau:

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

Giải các phương trình sau

a) \[\sqrt[3]{{1\,\,000x}} - \sqrt[3]{{64x}} - \sqrt[3]{{27x}} = 15\]                               b) \(\sqrt[3]{{x - 3}} + 3 = x\)

Thực hiện phép tính

a) \(\sqrt[3]{{\frac{1}{2}}} \cdot \sqrt[3]{{ - 18}} \cdot \sqrt[3]{3}\).                                    b) \(\sqrt[3]{{\left( {\sqrt 2  + 1} \right)\left( {3 + 2\sqrt 2 } \right)}};\;\;\sqrt[3]{{\left( {4 - 2\sqrt 3 } \right)\left( {\sqrt 3  - 1} \right)}}\).

c) \(\left( {\frac{1}{2}\sqrt[3]{2} - \frac{1}{4}\sqrt[3]{{16}}} \right) \cdot \sqrt[3]{4}\).                                              d)\[\left( {\frac{1}{2}\sqrt[3]{9} - 2\sqrt[3]{3} + 3\sqrt[3]{{\frac{1}{3}}}} \right):2\sqrt[3]{{\frac{1}{3}}}\].

e) \(\left( {\sqrt[3]{9} - \sqrt[3]{6} + \sqrt[3]{4}} \right)\left( {\sqrt[3]{3} + \sqrt[3]{2}} \right)\).

Thực hiện phép tính:

a)\[{\left( {\sqrt[3]{4} + 1} \right)^3} - {\left( {\sqrt[3]{4} - 1} \right)^3}\]                                                b)\[\left( {12\sqrt[3]{2} + \sqrt[3]{{16}} - 2\sqrt[3]{2}} \right)\left( {5\sqrt[3]{4} - 3\sqrt[3]{{\frac{1}{2}}}} \right)\].

c) \[\left( {12\sqrt[3]{2} + \sqrt[3]{{16}} - 2\sqrt[3]{2}} \right)\left( {5\sqrt[3]{4} - 3\sqrt[3]{{\frac{1}{2}}}} \right)\].