If $\displaystyle x^6-2=0$ then you can rarrange to get $\displaystyle x^6 = 2$. Now take the 6th root of both sides:
$\displaystyle (x^6)^{1/6} = 2^{1/6} $
$\displaystyle (x^6)^{1/6} = x^1 = x$. This comes from the rule that $\displaystyle ( x^a)^b = x^{(ab)}$. So you have $\displaystyle x = 2^{1/6}$ or $\displaystyle x = \sqrt[6] 2$.
Hence $\displaystyle \sqrt[6]2 $ is a solution to $\displaystyle x^6-2=0$.