Câu 22:
\(\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x\cdot\sqrt{x\cdot\sqrt{x\cdot x^{\dfrac{1}{2}}}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x\cdot\sqrt{x\cdot\sqrt{x^{\dfrac{3}{2}}}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x\cdot\sqrt{x\cdot x^{\dfrac{3}{4}}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x\cdot\sqrt{x^{\dfrac{5}{4}}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x\cdot x^{\dfrac{5}{8}}}:x^{\dfrac{11}{16}}\)
\(=\sqrt{x^{\dfrac{13}{8}}}:x^{\dfrac{11}{16}}=x^{\dfrac{13}{16}}:x^{\dfrac{11}{16}}=x^{\dfrac{1}{8}}\)
=>Chọn C
Câu 23:
\(\dfrac{1}{3}< \dfrac{1}{2}\)
=>\(a^{\dfrac{1}{3}}< a^{\dfrac{1}{2}}=\sqrt{a}\)(Vì a>1)
=>Loại C
\(a^{2018}< a^{2019}\)(do a>1)
=>\(\dfrac{1}{a^{2018}}>\dfrac{1}{a^{2019}}\)
=>Loại D
\(\dfrac{\sqrt[3]{a^2}}{a}=a^{\dfrac{2}{3}}:a=a^{-\dfrac{1}{3}}< 1\)
=>Loại B
=>Chọn A
Câu 21:
\(K=\sqrt[3]{\dfrac{2}{3}\cdot\sqrt[3]{\dfrac{2}{3}\cdot\sqrt{\dfrac{2}{3}}}}\)
\(=\sqrt[3]{\dfrac{2}{3}\cdot\sqrt[3]{\dfrac{2}{3}\cdot\left(\dfrac{2}{3}\right)^{\dfrac{1}{2}}}}\)
\(=\sqrt[3]{\dfrac{2}{3}\cdot\sqrt[3]{\left(\dfrac{2}{3}\right)^{\dfrac{3}{2}}}}\)
\(=\sqrt[3]{\dfrac{2}{3}\cdot\left(\dfrac{2}{3}\right)^{\dfrac{3}{2}\cdot\dfrac{1}{3}}}=\sqrt[3]{\dfrac{2}{3}\cdot\left(\dfrac{2}{3}\right)^{\dfrac{1}{2}}}\)
\(=\sqrt[3]{\left(\dfrac{2}{3}\right)^{\dfrac{3}{2}}}=\left(\dfrac{2}{3}\right)^{\dfrac{3}{2}\cdot\dfrac{1}{3}}=\left(\dfrac{2}{3}\right)^{\dfrac{1}{2}}=\sqrt{\dfrac{2}{3}}\)
=>Chọn B
Câu 20:
\(F=\dfrac{a^{\dfrac{2}{3}}\left(\sqrt[3]{a^{-2}}-\sqrt[3]{a}\right)}{a^{\dfrac{1}{8}}\cdot\left(\sqrt[8]{a^3}-\sqrt[8]{a^{-1}}\right)}\)
\(=\dfrac{a^{\dfrac{2}{3}}\cdot\left(a^{-\dfrac{2}{3}}-a^{\dfrac{1}{3}}\right)}{a^{\dfrac{1}{8}}\cdot\left(a^{\dfrac{3}{8}}-a^{-\dfrac{1}{8}}\right)}\)
\(=\dfrac{a^{\dfrac{2}{3}}\cdot a^{-\dfrac{2}{3}}-a^{\dfrac{2}{3}}\cdot a^{\dfrac{1}{3}}}{a^{\dfrac{1}{8}}\cdot a^{\dfrac{3}{8}}-a^{\dfrac{1}{8}}\cdot a^{-\dfrac{1}{8}}}\)
\(=\dfrac{1-a}{a^{\dfrac{1}{2}}-1}=\dfrac{1-a}{\sqrt{a}-1}=-\sqrt{a}-1\)
Thay \(a=2017^{2018}\) vào F, ta được:
\(F=-\sqrt{2017^{2018}}-1=-1-2017^{1009}\)
