Why is $a^n * a^m = a^{n+m}$, $a^n * b^n = (a*b)^n$ and NOT $a^n * b^m = (a*b)^{n+m}$??? Why is $a^n \times a^m = a^{n+m}$, and $a^n \times b^n = (a\times b)^n$; but NOT $a^n \times b^m = (a\times b)^{n+m}$?
I know the properties, but I still cannot be convinced! If the first two are true, why isn't the last one? In other words, why doesn't applying two correct steps at the same time provide a correct unique step?
 A: HINT
Do not focus on algebraic manipulation for a moment, but focus on the definition of power:
$$a^n=a \times a\times \ldots \times a\text{; for n times of }a$$
And you'll figure out why.
A: $$ a^n :=  \underbrace{ a \times a \times a   \times... \times a} \\ n $$
and :$$ a^m :=  \underbrace{ a \times a \times a   \times... \times a} \\ m $$
thus :
$$a^m \times a^n := \underbrace{ (a \times a \times... \times a)\times(a \times a \times... \times a)} =a^{n+m}\\ n+m$$
Now :
$$ a^n :=  \underbrace{ a \times a \times a   \times... \times a} \\ n $$
$$ b^n :=  \underbrace{ b \times b \times b   \times... \times b} \\ n $$
thus :
$$a^n \times b^n := \underbrace{ (ab \times ab \times ab \times...    \times ab)} =(ab)^{n}\\ n+m$$
A: For exploration purposes, let's see what rule we might get from the first two rules.  If $n\geq m$ (just to avoid negative exponents),
$$a^nb^m=a^{n-m}a^mb^m=a^{n-m}(ab)^m$$
(Be careful with what you mean by "two correct steps at the same time."  It's a reasonable conjecture that it might be the case that $a^nb^m=(ab)^{n+m}$, but this is certainly not what you get by doing both rules "at once."  The only rules you can apply at once are ones which can be performed in either order, for instance two additions in $(1+2)+(3+4)=3+7$, but in your case not only can you not perform the rules in either order to $a^nb^m$, but neither rule actually applies.)
