How does $2^n + 2^n = 2^{n+1}$? What property of exponents can be used to show that 
$$2^n + 2^n = 2^{n+1}$$
Does this work for all constants raised to a variable exponent?

I still remember trying to make my friends understand how we can take $2^n$ outside, when I was in middle school.
$$2^n + 2^n \\
= 2(2^{n-1} + 2^{n-1}) \\
= 2^2(2^{n-2} + 2^{n-2}) \\
= 2^3(2^{n-3} + 2^{n-3})\\
\dots\\ \dots
$$
$$=2^n(2^{n-n} + 2^{n-n})\\
= 2^n(2^0 + 2^0)\\
= 2^n(1 + 1)\\
= 2^n\cdot 2^1\\
= 2^{n+1}$$
But really, all they had to understand was that multiplication is repeated addition (something which they knew but didn't know how to apply):
$$a \times b =  \sum^a b = \underbrace{b + b + b + \dots}_{a \text{ times}} $$
So, naturally,
$$2^n + 2^n = 2\times 2^n = 2^{n+1}$$

Now, there are an infinite number of similar equations to this one you find interesting:
$$ 3^n + 3^n + 3^n = 3^{n +1}\\
4^n + 4^n + 4^n + 4^n = 4^{n+1}\\
5^n + 5^n + 5^n + 5^n + 5^n = 5^{n+1}\\
\dots \\ \dots$$

Generally,
$$ \sum^a a^n = a^{n+1} \quad \forall\space n>0$$

Please note, $a^n + a^n = a^{n+1} $ is only true if $a = 0 ,2$ and $n>0$
$$[\because 0^n + 0^n = 0 + 0 = 0 = 0^{ \text{potatoes} } = 0^{ \text{tomatoes} } = 0^{n+1}]$$
