What real numbers do algebraic numbers cover? Hardy and Wright mention ( though don't give a proof ) that any finite combination of real quadratic surds is an algebraic number. For example
$\sqrt{11+2\sqrt{7}}$. Are all finite combinations of cube root, fourth root ... $n^{th}$ root also algebraic ? such as $\sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}+\sqrt[9]{2}$.

The question in the title itself:

What real numbers do algebraic numbers cover?

has a rather amusing answer:

Practically none.

There are only countably many algebraic numbers, since each one is a root of some polynomial over $\mathbb{Q}$, and there are countably many such polynomials each of which has finitely many roots in $\mathbb{C}$. But there are uncountably many real numbers.
