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

Question

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

That is basic algebra. `2*(2^4) = 2*(2*2*2*2) = 2*2*2*2*2 = 2^5` — ypercubeᵀᴹ, Feb 21 '13 at 07:57

score 2 · Answer 1 · answered Feb 21 '13 at 08:02

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

answered Feb 21 '13 at 08:02

Pankrates

Ah yes in the last step, rule for multiplication of powers is applying, so simple ..... First bases are same but adding and then bases are same and multiplying ... thanks – Basit Feb 21 '13 at 08:08

paxdiablo · Accepted Answer · 2013-02-21T08:47:08.820

2

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

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

^{(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

edited Feb 21 '13 at 08:47

answered Feb 21 '13 at 08:02

paxdiablo

2 Answers2