TITLE: How does $2^{k+1} = 2 \times 2^k$?
QUESTION [9 upvotes]: 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.

REPLY [15 votes]: 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 $$x^k \times x = \underbrace{x \times x \times \cdots \times x}_{k \textrm{ times}} \times x =  \underbrace{x \times x \times \cdots \times x}_{k+1 \textrm{ times}}.$$