Question

If $x=2^{1/3}+2^{-1/3}$ prove that $2x^3-6x-5=0$

Answer 1

$x=2^{\frac{1}{3}}+2^{\frac{-1}{3}}$ ......$\left(1\right)$

Cubing both sides, we get

$x^3={(2^{\frac{1}{3}}+2^{\frac{-1}{3}})}^3$

$x^3=2+2^{-1}+3\times 2^{\frac{1}{3}}\times 2^{\frac{-1}{3}}(2^{\frac{1}{3}}+2^{\frac{-1}{3}})$

$x^3=2+2^{-1}+3\times 2^{\frac{1}{3}}\times 2^{\frac{-1}{3}}\left(x\right)$ from $\left(1\right)$

$x^3=2+\frac{1}{2}+3x$

$x^3=\frac{5}{2}+3x$

$2x^3-6x-5=0$

Hence proved.