Question
If $a=2^{\frac{1}{3}}-2^{\frac{-1}{3}}$, prove that $2 a^3+6 a=3$
✓
Answer
$a=2^{\frac{1}{3}}-2^{\frac{1}{3}} $
$ \Rightarrow a=2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}} $
$ \Rightarrow a^3=\left(2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}}\right)^3 $
$ =2-\frac{1}{2}-3\left(2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}}\right) $
$ \Rightarrow a^3=\frac{4-1}{2}-3 a $
$\Rightarrow a^3=\frac{3}{2}-3 a $
$\Rightarrow 2 a^3+6 a=3 .$
$ \Rightarrow a=2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}} $
$ \Rightarrow a^3=\left(2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}}\right)^3 $
$ =2-\frac{1}{2}-3\left(2^{\frac{1}{3}}-\frac{1}{2^{\frac{1}{3}}}\right) $
$ \Rightarrow a^3=\frac{4-1}{2}-3 a $
$\Rightarrow a^3=\frac{3}{2}-3 a $
$\Rightarrow 2 a^3+6 a=3 .$
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