Tính các căn bậc ba sau

a) \(\sqrt[3]{{64}}\)           b) \(\sqrt[3]{{ - 512}}\).                                            c) \(\sqrt[3]{{0,064}}\)

d) \(\sqrt[3]{{ - 0,216}}\)  e) \(\frac{{\sqrt[3]{{500}}}}{{\sqrt[3]{4}}} + \sqrt[3]{{12}} \cdot \sqrt[3]{{18}}\)                 f) \(\frac{{\sqrt[3]{{12}} \cdot \sqrt[3]{6}}}{{\sqrt[3]{{576}}}} - \frac{{\sqrt[3]{{32}}}}{{\sqrt[3]{4}}}\)

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

a) \(\sqrt[3]{{\frac{1}{2}}} \cdot \sqrt[3]{{ - 18}} \cdot \sqrt[3]{3} = \sqrt[3]{{\frac{1}{2} \cdot ( - 18) \cdot 3}} = \sqrt[3]{{ - 27}} =  - 3\).

b) \(\sqrt[3]{{\left( {\sqrt 2  + 1} \right)\left( {3 + 2\sqrt 2 } \right)}} = \sqrt[3]{{\left( {\sqrt 2  + 1} \right){{\left( {\sqrt 2  + 1} \right)}^2}}} = \sqrt[3]{{{{\left( {\sqrt 2  + 1} \right)}^3}}} = \sqrt 2  + 1\).

\(\sqrt[3]{{\left( {4 - 2\sqrt 3 } \right)\left( {\sqrt 3  - 1} \right)}} = \sqrt[3]{{{{\left( {\sqrt 3  - 1} \right)}^2}\left( {\sqrt 3  - 1} \right)}} = \sqrt[3]{{{{\left( {\sqrt 3  - 1} \right)}^3}}} = \sqrt 3  - 1\).

c) \(\left( {\frac{1}{2}\sqrt[3]{2} - \frac{1}{4}\sqrt[3]{{16}}} \right) \cdot \sqrt[3]{4} = \left( {\frac{1}{2}\sqrt[3]{2} - \frac{1}{2}\sqrt[3]{2}} \right) \cdot \sqrt[3]{4} = 0\).

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

e) \(\left( {\sqrt[3]{9} - \sqrt[3]{6} + \sqrt[3]{4}} \right)\left( {\sqrt[3]{3} + \sqrt[3]{2}} \right) = \left( {\sqrt[3]{{{3^2}}} - \sqrt[3]{{3 \cdot 2}} + \sqrt[3]{{{2^2}}}} \right)\left( {\sqrt[3]{3} + \sqrt[3]{2}} \right) = {\left( {\sqrt[3]{3}} \right)^3} + {\left( {\sqrt[3]{2}} \right)^3} = 3 + 2 = 5\).