Teaching non-integer exponents Say I'm teaching a younger student the concept of exponents. Now the basic example to begin with is say $2^3$ where it can be 'visualised' as having two blocks, then doubling the number of blocks, then doubling the number of blocks again because it can be expressed as $2\times2\times 2$.
My question is, how would you explain the concept of non-integer exponents to a younger student? What does it mean to raise something to the power of $\frac{3}{4}$?
Even worse is raising something to an irrational power.
Now this is where my main question is.
Say I wish to explain to a student at elementary school the graph of $y=2^x$ where they do understand the notion of rational exponents. But if they ask me about irrational exponents and the guarantee of the graph being continuous and smooth, how would you explain that? When I think about it more deeply, I realise that even I cannot really understand the continuity of the basic exponential curve because of the notion of irrational exponents.
 A: Although your question was mainly about irrational exponents, let me first answer for non-integer rational exponents, which you also mentioned, since these are the more important pedagogical problem. I can think of two approaches.
The first would be to study some problem with exponential growth, such as bacteria. Assume that at $t = 0$ you have one unit mass, and the amount is multiplied by 2 every hour. Make the further technical assumption that in equal periods of time, the amount is always multiplied by the same factor. This is a sensible assumption in the kind of problem I mentioned, so long as you have a very large number of individual organisms (and they have unlimited resources, etc.).
The first thing would be to establish that at $t = n$ hours ($n$ a positive integer) you have $2^n$ units. Next the same thing for $n$ a negative integer. Then ask the question for $t = 30$ minutes. This will motivate the definition of $2^{1/2}$, and you can make similar arguments for $2^{1/3}$, $2^{2/3}$, $2^{3/2}$, $2^{-3/4}$, etc.
The second approach is to see how $2^x$ must be defined for rational $x$ if we wish all the usual rules of exponents valid for integers to continue to hold, particularly $2^{x+y} = 2^x 2^y$. 
With respect to irrational exponents, I would focus on monotonicity rather than continuity. I would state (without proof of course, unless you're dealing with an exceptional student) that there is only one way to define the function $2^x$ for irrational exponents in such a way that the function remains increasing. 
What this boils down to for, say, $2^{\sqrt{2}}$, is that it is the unique number that is greater than $2^q$ whenever $q$ is a rational number less than $\sqrt{2}$, and less than $2^r$ whenever $r$ is a rational number greater than $\sqrt{2}$. In fact, I would probably explain it in this more concrete way rather than in terms of increasing functions.
So in practice, you're looking for a number that is more than $2^1$, $2^{1.4}, 2^{1.41}$, etc., and less than $2^2$, $2^{1.5}$, $2^{1.42}$, etc. This can turn into a fairly concrete exercise on a calculator, and it is plausible that there will be exactly one number that fits between these sequences. 
In order to prove things like this, you first have to have a precise definition of what a real number is, and the operations on real numbers such as addition and multiplication.
Continuity results from the theorem that says that an increasing, one-to-one and onto function between two open intervals is automatically continuous. In order to reach this point, you need to prove, above and beyond what I've already stated, that every positive number is of the form $2^x$ for some number $x$. A proof of this would use the least upper bound property of the real numbers. 
