Exponents show up in Big-O when algorithms branch or nest. means two nested loops over items. means three nested loops. These are polynomial and usually acceptable for small .
is exponential. It means the work doubles with every extra input element. At , that's about operations. At , about . At , you're waiting hours.
The growth rate hierarchy you should remember: . Polynomial beats exponential for large , which is why converting an brute force into an dynamic programming solution is such a big win.