a. \(\sqrt[3]{2x+1}=3\)
\(\Leftrightarrow2x+1=27\)
\(\Leftrightarrow x=13\)
b. \(\sqrt[3]{2-3x}=-2\)
\(\Leftrightarrow2-3x=-8\)
\(\Leftrightarrow x=\dfrac{10}{3}\)
c. \(\sqrt[3]{x+1}+1=x\)
\(\Leftrightarrow\sqrt[3]{x+1}=x-1\)
\(\Leftrightarrow x+1=x^3-3x^2+3x-1\)
\(\Leftrightarrow x^3-3x^2+2x-2=0\)
d. \(\sqrt[3]{x+1}+\sqrt[3]{7-x}=2\)
Đặt: \(\left\{{}\begin{matrix}a=\sqrt[3]{x+1}\\b=\sqrt[3]{7-x}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a^3=x+1\\b^3=7-x\end{matrix}\right.\)
Ta có: \(\left\{{}\begin{matrix}a+b=2\\a^3+b^3=8\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=2-b\\\left(2-b\right)^3+b^3=8\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=2-b\\8-12b+6b^2-b^3+b^3=8\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=2-b\\6b^2-12b=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=2-b\\\left[{}\begin{matrix}b=2\\b=0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=0\\b=2\end{matrix}\right.\\\left\{{}\begin{matrix}a=2\\b=0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+1=0\\7-x=8\end{matrix}\right.\\\left\{{}\begin{matrix}x+1=8\\7-x=0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow x=7\)
e. \(\sqrt[3]{x-2}+\sqrt{x+1}=3\)\(\left(x\ge-1\right)\)
Đặt: \(\left\{{}\begin{matrix}a=\sqrt[3]{x-2}\\b=\sqrt{x+1}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a^3=x-2\\b^2=x+1\end{matrix}\right.\)
Ta có: \(\left\{{}\begin{matrix}a+b=3\\a^3-b^2=-3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=3-b\\\left(3-b\right)^3-b^2=-3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=3-b\\27-27b+9b^2-b^3-b^2=-3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=3-b\\-b^3+8b^2-27b+30=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=3-b\\b=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x-2=1\\x+1=4\end{matrix}\right.\) \(\Leftrightarrow x=3\) (n)