How to add powers? 
$$5^3 + 5^3 + 5^3 + 5^3 + 5^3= 5^n$$ or $$5\times5^3=5^n$$ or $$125\times5=5^n$$
  What is $n$?

P.S.: I know how to multiply these powers but I've never known about how to add the same power (or multiply it by a number), to get a different power. 
I searched the question up, but all it comes up with is "how to simplify like terms". I know that $5^n$ equals $625$ but how can I work out what power that is? (It is $5^4$, but that's not the point of the question.)  
The question is, is there some index law that would help me solve this? If not, how do I solve this question anyways? I want some rule such as $a^x+a^y=a^z$, when $x, y, z$ are variables. 
Thanks.

Think about it as factoring $5^3$ from the left hand side, so we have:
$$5^3 + 5^3 + 5^3 + 5^3 + 5^3 = 5^3\left(1+1+1+1+1\right) = 5^3(5) = 5^4$$
