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

• The left side is $5\times 5^3$. – Mark Viola May 13 '17 at 23:57
• Hint: what is $x+x+x+x+x$? – lulu May 13 '17 at 23:57
• @Dr.MV That's why I said I never knew how to multiply a number by a power to get a different power. But I'll edit the question to show more working. Thanks – bio May 13 '17 at 23:59
• As Dr. MV wrote: the left side is $5^1 \times 5^3$. So....... – David G. Stork May 13 '17 at 23:59
• @bio It seems like it. If you want a rule that tells you $a^x + a^y = a^z$ when you have $a$, $x$, and $y$, then everyone here seems to be misunderstanding you. I don't believe there's an addition rule for powers in general, unless you do factoring tricks like in this specific case. – Axoren May 14 '17 at 0:10

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

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

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

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