Cho hàm số \(f\left( a \right)=\frac{{{a}^{\frac{-1}{3}}}\left( \sqrt[3]{a}-\sqrt[3]{{{a}^{4}}} \right)}{{{a}^{\frac{1}{8}}}\left( \sqrt[8]{{{a}^{3}}}-\sqrt[8]{{{a}^{-1}}} \right)}\) với \(a>0,a\ne 1.\) Tính giá trị \(M=f\left( {{2021}^{2020}} \right).\)

Ta có: \(f\left( a \right)=\frac{{{a}^{\frac{-1}{3}}}\left( \sqrt[3]{a}-\sqrt[3]{{{a}^{4}}} \right)}{{{a}^{\frac{1}{8}}}\left( \sqrt[8]{{{a}^{3}}}-\sqrt[8]{{{a}^{-1}}} \right)}=\frac{{{a}^{\frac{-1}{3}}}\left( {{a}^{\frac{1}{3}}}-{{a}^{\frac{4}{3}}} \right)}{{{a}^{\frac{1}{8}}}\left( {{a}^{\frac{3}{8}}}-{{a}^{\frac{-1}{8}}} \right)}=\frac{{{a}^{\frac{-1}{3}}}{{a}^{\frac{1}{3}}}\left( 1-a \right)}{{{a}^{\frac{1}{8}}}{{a}^{\frac{-1}{8}}}\left( {{a}^{\frac{1}{2}}}-1 \right)}=\frac{\left( 1-a \right)}{\sqrt{a}-1}=-\sqrt{a}-1\)

\(\Rightarrow f\left( {{2021}^{2020}} \right)=-{{\left( {{2021}^{2020}} \right)}^{\frac{1}{2}}}-1=-{{2021}^{1010}}-1.\)