How to evaluate $\sqrt[3]{a + ib} + \sqrt[3]{a - ib}$? The answer to a question I asked earlier today hinged on the fact that
$$\sqrt[3]{35 + 18i\sqrt{3}} + \sqrt[3]{35 - 18i\sqrt{3}} = 7$$
How does one evaluate such expressions? And, is there a way to evaluate  the general expression
$$\sqrt[3]{a + ib} + \sqrt[3]{a - ib}$$

The square of this is $2x+2\sqrt{x^2-y^2}$.  One could find this.
Regarding $\sqrt[3]{35 + 18i\sqrt{3}} + \sqrt[3]{35 - 18i\sqrt{3}} = 7$, the modulus of this number $35+18\sqrt{-3}$ is 2197, which is $13^3$.  This means that we are looking for a number whose cube is $35+18\sqrt{-3}$.  An example of such a number is $3.5 + 0.5\sqrt{-3}$, which we cube to get the expression of $35+18\sqrt{-3}$.  
The conjucate gives $3.5-0.5\sqrt{-3}$ gives a cube of $35-18\sqrt{-3}$.
The process here is to multiply the two numbers together, to establish the modulus.  Here, the product gives $2197$, a known cube.  In general, it needs to be a cube to fall on an integer real.  One could also divide the two numbers, in the form $\sqrt[3]{35 + 18i\sqrt{3}} / \sqrt[3]{35 - 18i\sqrt{3}}$, to get a value on the unit circle.  
However, it is easier to trace around a circle of radius $\sqrt{13}$, by looking for solutions to $x^2 + 3y^2 = 52$ for solutions of the type $(x+y\sqrt{-3})/2$.  One finds numbers like $7,3$, and then $5,3$ and $2,4$, all of which are likely suspects.  
It's a similar method that i have employed in more complex integer systems.
