Find the cardinality of a set of numbers which can be represented in form

Question

What is the cardinality of set of numbers which can be represented in form $\sqrt a+\sqrt[3]b + \pi\sqrt[4] c$, where $\ a, b, c \in \Bbb Q$.

I intuitively think, that the set is countable, but I can't prove it.

Answer 1

There is a clear surjection from $\mathbb Q^3$ to that set ( recall that $\mathbb Q^3$ is countable). and it is also clear that the set is not finite. So its cardinality is that of the natural numbers.

Answer 2

How many triples $(\sqrt a, \sqrt[3] b, \pi\sqrt[4] c$) are there with $a,b,c$ rational? It is countably infinite. Now sum the three components of the triples to get a number in the format of your question. Possibly two different triples lead to the same answer. So the set set of sums could be either finite or countable.

Now consider the proper subset of the triples with $(\sqrt{n^2}, 0,0)$ for positive integers $n$ ; the sums from this subcollection contains all positive integers. So the set is in fact countably infinite.