How this formula is deriving 2^N + 2^N = 2^(N+1) [closed]

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If the bases are same and variables are adding , then it becomes like this

^ represent power
X^N + X^N = 2X^N not X^2N

like we take common

X^N(1 + 1) = 2X^N

but in the case 2^N + 2^N = 2^(N+1)

if we take common

2^N(1 + 1) = (2)2^N

How it is becoming

2^(N+1)

I read this formula in a book Data Structures and algorithm analysis in Java 3rd edition. I am confuse.

Thanks

Answer 1

2n + 2n is equal to 2n * 2 or 2n * 21.

That's equivalent to 2n+1 because xm * xn = xm+n (see note 1 below).

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^{(note 1)} As to why this is the case, you can see the reason here:

x2 * x3
= (x * x) * (x * x * x)
= x * x * x * x * x
= x5

Answer 2

Here the rule for multiplication of powers is applied

X^N * X^N = X^(N+N)

So if we take your example now

2^N + 2^N = 2*(2^N) = 2^1 * 2^N = 2^(1+N) = 2^(N+1)

With extra steps for clarity