Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
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
up vote 15 down vote accepted

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

share|cite|improve this answer
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.

Also a look at may help.

share|cite|improve this answer
Nevermind, I'm just having a hard time wrapping my head around it. I'll just have to accept that it's true. – Unknown Oct 30 '12 at 20:03
You dont have to accept it, accepting is like believing, you can't be sure about it. If you really want to learn maths, you should question everything possible, that way you can really get behind a subject and understand what you are doing. Take a look at the wiki article, maybe it helps! – Stefan Oct 30 '12 at 20:07
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.