why is $X^n\cdot X^m = X^{n+m}$ I like to understand how things work. It's the only way I'll remember the rule. If anyone has a way of breaking this down into logical steps to get from one to the other... it would be greatly appreciated.
why is 
$X^n\cdot X^m = X^{n+m}$ ?
 A: If $n$ and $m$ are positive integers, then remember that $X^n$ is shorthand for 
$$ \underbrace{X\cdot X\cdots X}_{\text{$n$ factors}}$$
So $X^n\cdot X^m$ is:
$$X^n \cdot X^m = \underbrace{(X\cdot X\cdots X)}_{\text{$n$ factors}} \cdot \underbrace{(X\cdot X \cdots X)}_{\text{$m$ factors}} = \underbrace{X \cdot X\cdots X}_{\text{$n+m$ factors}} = X^{n+m}.$$
For negative integers, remember that if $n\gt 0$, then $X^{-n} = \frac{1}{X^n}$. So we have that 
$$X^{-n}\cdot X^{-m} = \frac{1}{X^n}\cdot \frac{1}{X^m} = \frac{1}{X^n\cdot X^m} = \frac{1}{X^{n+m}} = X^{-(n+m)} = X^{-n-m}.$$
From there, using the other exponent rules, you can also get that it works for exponents that are fractions; and then once you define arbitrary exponents, for any exponent.
A: $X^n$ is $X$ multiplied by itself $n$ times. $X^m$ is $X$ multiplied by itself $m$ times. When you multiply these together, you'll get $X$ multiplied by itself $n+m$ times, which is $X^{n+m}$.
Here's an example with $n=2$ and $m=3$: $$X^2 \cdot X^3 = (X \cdot X) \cdot (X \cdot X \cdot X) = X^5.$$
