What are the roots of quintics? I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general polynomial of degree greater than four.
When I typed such a polynomial into wolfram alpha, it gave me decimal approximations of roots, and that was the only type of expression there was.
My question is, do we have any idea what how we can express a general root other than via decimal approximations?

You ask for a general method other than via decimal approximation. Because of Galois, such a method must be  transcendental, as opposed to algebraic (which you call "arithmetic"). For instance, by analyzing the symmetries of the icosahedron, Felix Klein showed that the solutions of the quintic could be expressed by taking radicals of values of hypergeometric functions (the formulae are analogous to those giving the algebraic solutions of the cubic).  A variant is explained in every detail in the memoir  "Résolution de l'équation du cinquième degré" by P. Gabriel & S. Benzekry, www.ann.jussieu.fr/~gabriel/documents/Memoire.pdf. The main steps are:
 - by using the Newton relations between the coefficients and the roots, transform the general quintic into a "principal form" $X^5 + 5aX^2 + 5bX + c$
- by using the symmetries of a polyhedron inscribed in the Riemann sphere, transform this into the so called Brioschi equation, which is another particular quintic
- transform this in turn into the Jacobi equation, which is a sextic whose coefficients involve the classical Weierstrass functions \Delta and $g_2$
- solve the Jacobi equation by bringing it back to a cubic                            
