# How does $2^{k+1} = 2 \times 2^k$?

I'm trying to learn mathematics by Induction but my knowledge of simplifying algebraic equations is crippling me.

Thanks.

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When you don't understand generalized formulas, expressions, etc., try making sense of them by using concrete examples. For instance $2 \cdot 2^2 = 2 \cdot 4 = 8 = 2^3 = 2^{2+1}$, etc. Then you can grasp what's happening or why the generalization makes sense. Also, Stefan's link to exponentiation below may help. –  amWhy Oct 30 '12 at 20:09
Someone should probably mention that the formula you are asking about is usually taken to be the formal definition of integer exponentiation (it is a definition by recursion.) So if you are having trouble finding a formal proof of it, that's why. –  Trevor Wilson Oct 30 '12 at 20:56

By the rules of exponentiation,

$x^{k} \times x = x^{k+1}$.

If $k$ is an integer, $x^k = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}}.$

So $$x^k \times x = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}} \times x = \underbrace{x \times x \times \cdots \times x}_{k+1 \textrm{ times}}.$$

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unless $x$ is a Grassmann number ;) –  Valentin Oct 30 '12 at 21:20
$2^{k+1}$ is $2$ multiplied with itself k+1 times. $2\cdot2^k$ is $2$ multiplied $k$ times with itself and an additional $2$ makes it multiplied $k+1$ times with itself.
Oh, it makes sense. I was confused, but it is quite simple. Since multiplying 2 by $2^k$ is just like increasing k by 1. Very easy stuff. Thanks. –  Unknown Oct 30 '12 at 20:16