Cube or root or square or fourth?

$\text {Let } a = \sqrt [3] {x}$

Then we are looking for the value of:

$S = a + \frac {1}{a}$

Our equation can be written as:

$\sqrt [3] { 3(a - \frac {1}{a}) } = 2$

After cubing both sides and dividing by 3, we get:

$a - \frac {1}{a} = \frac {8}{3}$

Let's square both sides:

$a^2 - 2 + \frac {1}{a^2} = \frac {64}{9}$

If we add 4 to both sides:

$a^2 + 2 + \frac {1}{a^2} = \frac {100}{9}$

$(a + \frac {1}{a} )^2 = \frac {100}{9}$

$a + \frac {1}{a} = ± \frac {10}{3}$

Hence, our answer should be:

$\boxed { \text {Both } \frac {10}{3} \text { and } - \frac {10}{3} }$