The Cube Root of ii?

First, you can read about the definition of ii, Imaginary Unit in the Algebra Glossary


\struti3=a+ib\sqrt[3]{\strut i}=\displaystyle \frac{\sqrt{a}+i}{b}

where aa and bb is positive integers and aa is not a square number.

What is the value of a+ba+b?

This problem is inspired by The Square Root of ii?


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