Great question! To begin with, if this were an *equation* and not an inequality, you'd e totally fine to set the bases to be whatever identical things you wanted. The restrictions here come from the fact that we're dealing with an inequality, and in other areas, too, inequalities require more "caution" than do equations. (Think, for example, of how when you multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign to make the statement remain true.)

So, let's say we make our base any number x between (but exclusive of) 0 and 1. The unique thing about that region on the number line is that squaring, or cubing, or "to-the-fourh"ing (etc.) a number yields a result that is LESS than the original number. So for our x: 0<x<1, we'll actually have x^1>x^2>x^3>x^4 and so on. But taking the base away here would yield 1>2>3>4, which is clearly not true.

You also want to watch out for bases that are (or may be) negative numbers, since they, too, violate the "normal" pattern that raising a number to a bigger power yields a bigger number. For instance, (-10)^2>(-10)^3, since the square is positive and the cube negative. But here again, stripping away the identical bases will be a flawed approach, because it's not true that 2>3.

So, to summarize, if you use the approach of equating the bases in an inequality, you need to stick to bases that are > 1 (because that's the range in which we can trust numbers to behave "normally" when we raise them to powers). If you use this approach in an equation, you needn't worry -- you can choose any base you like.

yikes! Those are great examples Ashley and they strengthen my awareness and concern. Thanks!