\[\begin{array}{l}
C = {\left( {2{a^{ - \frac{2}{3}}} - \frac{1}{2}{a^{\frac{3}{2}}}} \right)^3} = {\left( {\frac{2}{{{a^{\frac{2}{3}}}}} - \frac{1}{2}{a^{\frac{3}{2}}}} \right)^3} = {\left( {\frac{2}{{\sqrt[3]{{{a^2}}}}} - \frac{{\sqrt {{a^3}} }}{2}} \right)^3}\\
= {\left( {\frac{2}{{\sqrt[3]{{{a^2}}}}}} \right)^3} - 3.{\left( {\frac{2}{{\sqrt[3]{{{a^2}}}}}} \right)^2}.\frac{{\sqrt {{a^3}} }}{2} + 3.\frac{2}{{\sqrt[3]{{{a^2}}}}}.{\left( {\frac{{\sqrt {{a^3}} }}{2}} \right)^2} - {\left( {\frac{{\sqrt {{a^3}} }}{2}} \right)^3}\\
= \frac{8}{{{a^2}}} - .\frac{{12}}{{\sqrt[3]{{{a^4}}}}}.\frac{{\sqrt {{a^3}} }}{2} + \frac{6}{{\sqrt[3]{{{a^2}}}}}.\frac{{{a^3}}}{4} - \frac{{{{\left( {\sqrt {{a^3}} } \right)}^3}}}{8}\\
= \frac{8}{{{a^2}}} - 6.{a^{\frac{3}{2} - \frac{4}{3}}} + \frac{3}{2}.{a^{\frac{3}{2} - \frac{2}{3}}} - \frac{{{a^3}}}{8}\\
= \frac{8}{{{a^2}}} - \frac{{{a^3}}}{8} - 6{a^{\frac{1}{6}}} + \frac{3}{2}{a^{\frac{7}{3}}}.
\end{array}\]
