It's always a little bit tricky and annoying to write math problems on a GMAT blog (there's no way to comfortably write equations, unless you want to import images), but I'll do my best with this topic. I just want to make a few little comments about exponents, and about the ways in which the GMAT Official Guides (including the GMAT Quantitative Review Guide) can lull a perfectly good GMAT student into complacency on the topic. Hopefully, you know all of the basic exponent rules. If you're multiplying two terms with the same base, you'll add the exponents (example: x^3*x^4 = x^7). If you're dividing, you know that you subtract the exponents, and then you might encounter something like a "power of a power" question (example: (x^6)^2 = x^12). If you know these rules and a few other basics (i.e. what happens if an exponent is negative, a fraction, or zero), you'll be fine on the GMAT. Right?
If you're focused on the GMAT Official Guide questions... well, yes, you'll be mostly fine. Let's take a quick survey of the exponent questions in the problem solving section of the 12th edition of the GMAT Official Guide: #15, #28, #46, and #108 all contain exponents, but they're mostly a matter of calculating (or simplifying) some numbers. #104 and #110 look like exponent problems, but both are really about factors, not exponent properties. #137 could be solved using exponent properties, but is just as easily done with some simple calculations and logic...
I could go on. My point is, most of what you encounter in the GMAT Official Guides doesn't require much knowledge of exponent properties. And when you do need to use exponent properties, they're just covering the basics.
Funny, I didn't see anything basic last time I took the GMAT, and neither did most of my students who scored above 600. I keep hearing the same refrain: the exponent problems looked nothing like they do in the GMAT Official Guides.
I'm convinced--based on the GMATPrep, GMAT Focus, and the real thing--that the GMAT is much more likely to show you an exponent question that has something to do with factoring and/or "base conversion." Neither of these topics necessarily receive the attention they deserve in GMAT test-prep books.
Please accept my profuse apologies for the crappy notation, but here are a couple of examples of realistic, harder exponent problems (NOT from official GMAT materials, lest I incur the wrath of some bigshot NYC lawyer sent by the bigshots who write the GMAT):
(7^10 - 7^8)/3 = (2^x)(7^y). If x and y are integers, then what is the value of x + y?
3^(x-1) - 3^(x+1) = -(9^5)(2^3). What is the value of x?
I'm not going to post solutions until I'm begged repeatedly, but hopefully you see where I'm going with this. You're going to see GMAT exponent problems that require some factoring, as well as the ability to make some ostensibly unlikely connections. Unless you've seen these problems, you might be wondering where the heck the 2's are coming from.
Welcome to the GMAT. If you're not uncomfortable, you're probably not doing well. Either that, or you're way smarter than I am, and you shouldn't be wasting your time reading a blog posting about exponents on the GMAT.