Think of the power series ^2, ^3 etc as defining the characteristics of the <em>line on a graph</em> that the power function draws.
The awful reality is that the logical answer is buried in the axioms of set theory. So instead of having to teach kids axioms and derivations, just draw the lines on a graph as i described.
^0 = a line with zero slope, ^1 = a straight line with a slope of 1, ^2 = an exponential line... etc...
For kids, relating the power series to the shapes of lines on a graph will also help them later on when they learn about functions etc (like y = mx + c). Being able to associate the different powers with actual shapes on a graph will also help them to visualize relationships in physics, should they take that path. It's not the real truth, but a nice correlation with it's own merits.
