How to add powers?

Question

$$5^3 + 5^3 + 5^3 + 5^3 + 5^3= 5^n$$ 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.

Answer 1

Think about it as factoring $5^3$ from the left hand side, so we have:

$$5^3 + 5^3 + 5^3 + 5^3 + 5^3 = 5^3\left(1+1+1+1+1\right) = 5^3(5) = 5^4$$

Answer 2

In reply to your question at the end: the best you can say, in general, is that $$a^x+a^y=a^x(1+a^{y-x})$$

Answer 3

Since repeated addition is simply multiplication, you can define

$\ 5^3 + 5^3 + 5^3$

as

$\ 3(5^3) $

3 can be defined as $\ 5^x $ where $\ x = log$₅ 3

since powers that are multiplied are added together the expression becomes

5 ^{3 + log₅ 3}

Therefore $\ c*a^b = a^n$ where $\ n = b + log$ _ac