Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$.
Does it hold in general that every irrational number can be written in terms, i.e., in a closed form expression, of finitely many rational numbers?
For irrational numbers like $e$ and $\pi$ this is not immediately clear.
Best Answer
A little extra detail, you have included a function, square root. Suppose we add in a finite vocabulary of functions that can also be used, logarithm base $e,$ exponential, trigonometry, inverse trig, hyperbolic trig, your favorite list of less "elementary" functions suh as hypergeometric, Lambert W, whatever.
Any expression is still a finite string combining rational numbers and a fixed alphabet of functions. As a result, the outcome is a countable list of numbers. So, still an uncountable set not accounted for.