Are 2^n and n*2^n in the same time complexity?

You will have to go to the formal definition of the big O (O) in order to answer this question.

The definition is that f(x) belongs to O(g(x)) if and only if the limit limsupx → ∞ (f(x)/g(x)) exists i.e. is not infinity. In short this means that there exists a constant M, such that value of f(x)/g(x) is never greater than M.

In the case of your question let f(n) = n ⋅ 2n and let g(n) = 2n. Then f(n)/g(n) is n which will still grow infinitely. Therefore f(n) does not belong to O(g(n)).

A quick way to see that n⋅2ⁿ is bigger is to make a change of variable. Let m = 2ⁿ. Then n⋅2ⁿ = ( log₂m )⋅m (taking the base-2 logarithm on both sides of m = 2ⁿ gives n = log₂m ), and you can easily show that m log₂m grows faster than m.

I agree that n⋅2ⁿ is not in O(2ⁿ), but I thought it should be more explicit since the limit superior usage doesn't always hold.

By the formal definition of Big-O: f(n) is in O(g(n)) if there exist constants c > 0 and n₀ ≥ 0 such that for all n ≥ n₀ we have f(n) ≤ c⋅g(n). It can easily be shown that no such constants exist for f(n) = n⋅2ⁿ and g(n) = 2ⁿ. However, it can be shown that g(n) is in O(f(n)).

In other words, n⋅2ⁿ is lower bounded by 2ⁿ. This is intuitive. Although they are both exponential and thus are equally unlikely to be used in most practical circumstances, we cannot say they are of the same order because 2ⁿ necessarily grows slower than n⋅2ⁿ.

I do not argue with other answers that say that n⋅2ⁿ grows faster than 2ⁿ. But n⋅2ⁿ grows is still only exponential.

When we talk about algorithms, we often say that time complexity grows is exponential. So, we consider to be 2ⁿ, 3ⁿ, eⁿ, 2.000001ⁿ, or our n⋅2ⁿ to be same group of complexity with exponential grows.

To give it a bit mathematical sense, we consider a function f(x) to grow (not faster than) exponentially if exists such constant c > 1, that f(x) = O(cx).

For n⋅2ⁿ the constant c can be any number greater than 2, let's take 3. Then:

n⋅2ⁿ / 3ⁿ = n ⋅ (2/3)ⁿ and this is less than 1 for any n.

So 2ⁿ grows slower than n⋅2ⁿ, the last in turn grows slower than 2.000001ⁿ. But all three of them grow exponentially.

You asked "is the second in the same order as the first? Does the additional n multiplication there matter?" These are two different questions with two different answers.

n 2^n grows asymptotically faster than 2^n. That's that question answered.

But you could ask "if algorithm A takes 2^n nanoseconds, and algorithm B takes n 2^n nanoseconds, what is the biggest n where I can find a solution in a second / minute / hour / day / month / year? And the answers are n = 29/35/41/46/51/54 vs. 25/30/36/40/45/49. Not much difference in practice.

The size of the biggest problem that can be solved in time T is O (ln T) in both cases.