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Câu hỏi: Cho hàm số $f\left( a \right)=\dfrac{{{a}^{\dfrac{-1}{3}}}\left( \sqrt[3]{a}-\sqrt[3]{{{a}^{4}}} \right)}{{{a}^{\dfrac{1}{8}}}\left( \sqrt[8]{{{a}^{3}}}-\sqrt[8]{{{a}^{-1}}} \right)}$ với $a>0,a\ne 1.$ Tính giá trị $M=f\left( {{2021}^{2020}} \right).$
A. $M=1-{{2021}^{2020}}$
B. $M={{2021}^{1010}}-1.$
C. $M={{2021}^{2020}}-1.$
D. $M=-{{2021}^{1010}}-1.$
A. $M=1-{{2021}^{2020}}$
B. $M={{2021}^{1010}}-1.$
C. $M={{2021}^{2020}}-1.$
D. $M=-{{2021}^{1010}}-1.$
Ta có: $f\left( a \right)=\dfrac{{{a}^{\dfrac{-1}{3}}}\left( \sqrt[3]{a}-\sqrt[3]{{{a}^{4}}} \right)}{{{a}^{\dfrac{1}{8}}}\left( \sqrt[8]{{{a}^{3}}}-\sqrt[8]{{{a}^{-1}}} \right)}=\dfrac{{{a}^{\dfrac{-1}{3}}}\left( {{a}^{\dfrac{1}{3}}}-{{a}^{\dfrac{4}{3}}} \right)}{{{a}^{\dfrac{1}{8}}}\left( {{a}^{\dfrac{3}{8}}}-{{a}^{\dfrac{-1}{8}}} \right)}=\dfrac{{{a}^{\dfrac{-1}{3}}}{{a}^{\dfrac{1}{3}}}\left( 1-a \right)}{{{a}^{\dfrac{1}{8}}}{{a}^{\dfrac{-1}{8}}}\left( {{a}^{\dfrac{1}{2}}}-1 \right)}=\dfrac{\left( 1-a \right)}{\sqrt{a}-1}=-\sqrt{a}-1$
$\Rightarrow f\left( {{2021}^{2020}} \right)=-{{\left( {{2021}^{2020}} \right)}^{\dfrac{1}{2}}}-1=-{{2021}^{1010}}-1.$
$\Rightarrow f\left( {{2021}^{2020}} \right)=-{{\left( {{2021}^{2020}} \right)}^{\dfrac{1}{2}}}-1=-{{2021}^{1010}}-1.$
Đáp án D.