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

Question

I ask only because my textbook infers this in an example. Where should I go to learn more about this?

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

Thanks.

Answer 1

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

Answer 2

$2^{k+1}$ 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.

Also a look at http://en.wikipedia.org/wiki/Exponentiation may help.