How do you simplify exponent properties?

There are a few properties that will take a little while to learn, so reading this once will not necessarily teach you what you need to know.
1) exponent property of multiplication:
a^b(a^c)=a^b+c; what this means is if you have two terms with the same base multiplied by each other, you can add their exponents. Say that I have 2(2), we know that any singular number is raised to the first power. Both twos are raised to the first power so 2(2) is also 2^1+1, or 2^2. Similarly 2^2(2^3)=2^2+3=2^5. However if the terms have different bases this rule DOES NOT APPLY. (Sorry for caps, I needed emphasis).
2) exponent property of division:
a^b/a^c=a^b-c; what this means is that if you have two terms with the same base divided by each other you can subtract their exponents. For example 2^3/2^2=2^3-2=2^1=2.
3) exponent property of um... craziness: a^b^c=a^b(c); for this rule of exponents b and c don’t have to be the same. For example, if we have 2^2^3, 2^2^3=2^2(3)=2^6.
There are a few more, so these take patience, time, and attention to detail. I hope this helps a bit.