You can factor $x^1+x^2$ into $x(1+x)$, but it is not a power. There is no addition law for powers in the way you posit because positive integer powers equal repeated multiplication, not addition:
$$x^{n+m}=\underbrace{x\cdot x\cdot x\cdots x}_{n+m}=\underbrace{x\cdot x\cdots x}_n\cdot\underbrace{x\cdot x\cdots x}_m=x^{n}\cdot x^m.$$
However, the addition law would work for multiplication by integers, rather than integer powers:
$$(n+m)x=\underbrace{x+ x+ x\cdots x}_{n+m}=\underbrace{x+x\cdots x}_n \,+\, \underbrace{x+x\cdots x}_m =nx+mx.$$
Again, with $n+m$ a positive integer. There are conceptual quandries about repeated operations here that I will not go into, but suffice it to say these repetition formulas hold as valuable cases.