Is there a name for this type of expression? Forgive me if this seems like a silly question. I know that the following expression is an example of a polynomial: $a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ but I am wondering if there is a name for one that takes this form: $a_{4}\sqrt[4]{x}+a_{3}\sqrt[3]{x}+a_{2}\sqrt{x}+a_{1}x+a_{0}$.
 A: There's no name in common use for the set of expressions in exactly the form you quote. One problem with the concept is that they are not closed under multiplication, like polynomials are. For example we have
$$ (a_2\sqrt x + a_1x)(b_2\sqrt x + b_1 x) = a_2b_2 x + (a_1b_2+a_2b_1) x^{3/2} + a_1b_1 x^2 $$
where the middle term looks neither like $ax^n$ nor $a\sqrt[n]x$.
We can speak of things that look like general polynomials, but where we allow arbitrary (nonnegative) rational exponents rather than just natural numbers. This would be closed under multiplication and addition, and include your linear combinations of higher roots of $x$ as a special case.
But this generalized concept still doesn't seem to have a commonly used name of its own. As Robert Israel suggests, they can be described as "finite Puiseux series", but googling shows that this is a quite rare term. If you want to be understood, saying "polynomials in $\sqrt[n]x$ for some $n$" instead would probably be advisable.
A: You could call it a Puiseux polynomial.
A: It is a polynomial in $y=x^{\frac 1{12}}$ namely $$p(y)=a_4y^3+a_3y^4+a_2y^6+a_1y^{12}+a_0$$and is thus an element of the polynomial ring obtained by adjoining a $12^{th}$ root of $x$ to the original polynomial ring.
