Rules regarding exponents

Question

Given the following algebra problem:

$$2^{n+1}-1+2^{n+1}=2^{n+1+1}-1$$

I know $2^{n+1}=2^n2^1$ but just to confirm the truth of the problem above, I just assumed the left hand side is $2^{n+2}-1$ since that is what the right side is.

How do I add $2^{n+1}+2^{n+1}$?

Algebraically, what is the rule that explains the addition of exponents?

Sorry if this is a silly question, but I just would like to understand this small part from larger more complicated problem I was working on.

Answer 1

You have $2$ terms of $2^{n+1}$, meaning you have $$2\cdot 2^{n+1} -1 = 2^{(n+1)+1} - 1 = 2^{n+2}-1$$

Answer 2

Note that

$$2^{n+1}+2^{n+1}=2\cdot 2^{n+1}=2^1\cdot 2^{n+1}=2^{n+1+1}=2^{n+2}.$$