I think this is asking the same question as here except that I don't quite understand the accepted answer. For instance, I can produce(*) what I think is a counterexample:
$$\sqrt[3]{26+15\sqrt{3}} + \sqrt[3]{26-15\sqrt{3}}$$
I do not follow how, but it does seem to equal the integer 4. I do not see how this could be rewritten in the simple form in the linked question.
How can these two irrational numbers add up to a rational number? I see there are two parts that could cancel out if they weren't under the cube root, but the fact that they are throws a monkey wrench into it with regards to me figuring it out. (I'm also curious if there are other numbers like this - I tried different symmetrical numbers under the cube roots and different roots besides cubed but I couldn't find any others.)
(*) By produce, I mean I encountered this counterexample, went searching for how it could be, and found the linked question and didn't see it mentioned there.