Question

Rút gọn biểu thức sau:

\(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\)

 

Answer 1

Đặt \(x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}.\)

\(\Rightarrow x^3=\sqrt{5}+2-3\sqrt[3]{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}\left(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\right)-\sqrt{5}+2\)

         \(=4-3\sqrt[3]{5-4}.x\)( Vì \(x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\))

        \(=4-3x\)

\(\Rightarrow x^3+3x-4=0\Leftrightarrow\left(x^3-1\right)+\left(3x-3\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x+4\right)=0\Leftrightarrow x-1=0\)( Vì \(x^2+x+4=\left(x+\frac{1}{2}\right)^2+\frac{15}{4}>0\))

\(\Leftrightarrow x=1\)hay \(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=1\)