I have found that the most successful way to get my Algebra students to really learn the rules of exponents is to have them write out the problems according to the definition of an exponent (so x^2 = x * x, x^3 = x*x*x, etc.) and then simplify the resulting expressions.
This worksheet will help students discover all of the basic rules of exponents: when to add the exponents, when to subtract them, and when to multiply them. The challenging part for students is translating their observations (e.g. "add the exponents") into a formal algebraic rule (x^a*x^b=x^(a+b)), but that's where you come in: I assign this in groups of 4, and I ask them to call me over when they've discovered a new rule, and they get very excited when the find it!
I do a "mini-lesson" of only three examples before I set them to task: I work out an example similar to problems 1, 12, and 27, which covers the use of the definition of an exponent, how to handle exponents nested in parentheses, and how to handle a problem in which the denominator has more factors than the numerator.
Once they begin, I ask them to check answers with one another as the "slowest-pace" student finishes them. (I level with them, and tell them it's uncommon for everyone to finish at the same time. Often, it's the slower, methodical approach that yields more accurate results). I tell them if they have a question, to call me over, with the caveat that if Marcy has a question, I will ask Timmy, "Timmy, what is Marcy's question?", which forces them to ask one another for help and to assist one another before they need me.
As they finish Part A, I check in with them and ask if their group members caught one another's mistakes, and virtually all of them report that their group caught errors in their work.
It is intended that students will do these problems on their own paper.
The preview file is a pdf so you may see exactly what's included; the product file is a word document, which you may edit for your students.