Cho hàm số f(a)=a^-1/3( căn bậc ba a- căn bậc ba a^4) / a ^1/8(căn bậc 8 a^3- căn bậc tám a^-1) 

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 $M=f\left({2021}^{2020}\right).$

Trả lời

Sử dụng công thức $\sqrt[m]{a^n}=a^{\frac{n}{m}},a^m.a^n=a^{m+n}.$

Cách giải:

$\begin{array}{l}f\left(a\right)=\frac{a^{-\frac{1}{3}}\left(\sqrt[3]{a}-\sqrt[3]{a^4}\right)}{a^{\frac{1}{8}}\left(\sqrt[8]{a^3}-\sqrt[8]{a^{-1}}\right)}=\frac{a^{\frac{1}{3}}\left(a^{\frac{1}{3}}-a^{\frac{4}{3}}\right)}{a^{\frac{1}{8}}\left(a^{\frac{3}{8}}-a^{\frac{1}{8}}\right)}\\ =\frac{1-a}{a^{\frac{1}{2}}-1}=\frac{1-a}{\sqrt{a}-1}=-\frac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}-1}=-\sqrt{a}-1\\ \Rightarrow f\left({2021}^{2020}\right)=-\sqrt{{2021}^{2020}}-1=-{2021}^{1010}-1.\end{array}$

Chọn B.