I don't really under stand a simply and easy way to divide exponents without actually showing all the work. FOr example the problem 15^30/45^15 how could u find this answer this out actaully multipling the whole thing out does anything cancel each other out cause i don't understand how to use the powers laws because the bases and the exponents are both different??

Well, by commutativity of multiplication, if ab = c, then axbx = cx. Thus, you can say 4515 = 1515315 and the problem reduces to 1515/315 which can be further reduced to 515. We could have obtained this directly by noting 1530 = 1515315515 = 4515515.

Hope this helps,
Dave

The key is being able to switch your view of the problem from "whole" to "a bunch of parts put together." This is common throughout mathematics.

Don't just look at 15^30 as some huge, monolithic number. Keep it broken up into pieces you can understand.

When you see division you should think "Factor it" and/or "multiply top and bottom by the same thing"